Introduction to Robotics - Unistraeavr.u-strasbg.fr/~bernard/education/houston/slides_houston... · Introduction to Robotics Bernard BAYLE Joint course University of Strasbourg, University

Outline

Introduction to Robotics

Bernard BAYLE

Joint courseUniversity of Strasbourg, University of Houston, Telecom Paris Tech

2008–2009

Outline

Outline

1 Rigid-Body MechanicsMathematical BackgroundRotationRigid-Body TransformationRigid-Body Motion

2 Description of Robotic ManipulatorsKinematic ChainsModified Denavit-Hartenberg ParameterizationGeometric EquationsVelocity Equations

3 Robotic Manipulators ModellingConfiguration and Pose of a Robotic ManipulatorDirect KinematicsInverse KinematicsDifferential Kinematics

Rigid-Body Mechanics Description of Robotic Manipulators Robotic Manipulators Modelling

Points

NotationsF = (O, x , y , z) is a Cartesian right-hand orthonormal frame,with Gibbs convention.

Point M position : vector m with coordinates ∈ R3 :

m =

mxmymz

Point Motion m(t) : parametric curve in R3

Point Path m(s) : geometric path associated to the motion(s ∈ [0 1] normalized curvilinear abscissa)


Points



m =

mxmymz




Points



m =

mxmymz




Points



m =

mxmymz




Rigid-Body and Rigid-Body Transformation

Rigid body : for any pair of points with coordinates m andn :

||m(t)− n(t)|| = ||m(0)− n(0)|| = constant

Rigid-body pose : position and orientation of a frameattached to this rigid body in FRigid-body transformation : result of a rigid motion.=Application which converts the coordinates of any point ofthe rigid body from their initial to their final values.Application = rigid-body transformation ? If and only if itpreserves distances and orientations.

ConsequenceIn a rigid-body tranformation, a right-hand orthonormal frame ischanged into another right-hand orthonormal frame.




||m(t)− n(t)|| = ||m(0)− n(0)|| = constant

Rigid-body pose : position and orientation of a frameattached to this rigid body in F

Rigid-body transformation : result of a rigid motion.=Application which converts the coordinates of any point ofthe rigid body from their initial to their final values.Application = rigid-body transformation ? If and only if itpreserves distances and orientations.





||m(t)− n(t)|| = ||m(0)− n(0)|| = constant

Rigid-body pose : position and orientation of a frameattached to this rigid body in FRigid-body transformation : result of a rigid motion.

=Application which converts the coordinates of any point ofthe rigid body from their initial to their final values.Application = rigid-body transformation ? If and only if itpreserves distances and orientations.





||m(t)− n(t)|| = ||m(0)− n(0)|| = constant

Rigid-body pose : position and orientation of a frameattached to this rigid body in FRigid-body transformation : result of a rigid motion.=Application which converts the coordinates of any point ofthe rigid body from their initial to their final values.

Application = rigid-body transformation ? If and only if itpreserves distances and orientations.





||m(t)− n(t)|| = ||m(0)− n(0)|| = constant






||m(t)− n(t)|| = ||m(0)− n(0)|| = constant




Rotation matrix

NotationsF ′ = (O, x ′, y ′, z ′) right-hand orthonormal frame

x ′, y ′, z ′ : coordinates of x ′, y ′ and z ′ in F :

x ′ =

0@x ′.xx ′.yx ′.z

1A , y ′ =

0@y ′.xy ′.yy ′.z

1A and z′ =

0@z′.xz′.yz′.z

1A .

DefinitionR = (x ′ y ′ z ′) of dimension 3× 3 is the rotation matrix fromframe F to frame F ′ . . . or also the change of basis matrix.


Rotation matrix

NotationsF ′ = (O, x ′, y ′, z ′) right-hand orthonormal frame

x ′, y ′, z ′ : coordinates of x ′, y ′ and z ′ in F :

x ′ =

0@x ′.xx ′.yx ′.z

1A , y ′ =

0@y ′.xy ′.yy ′.z

1A and z′ =

0@z′.xz′.yz′.z

1A .

DefinitionR = (x ′ y ′ z ′) of dimension 3× 3 is the rotation matrix fromframe F to frame F ′ . . . or also the change of basis matrix.


Rotation matrix

Role of the rotation matrix

gives a representation of the rotation of a frame attachedto a rigid body, from F to F ′

allows to calculate the coordinates of a point in a newframe


Rotation matrix

Role of the rotation matrixgives a representation of the rotation of a frame attachedto a rigid body, from F to F ′


O

z′ z

y

Mx

y ′

x ′


Rotation matrix

Role of the rotation matrixgives a representation of the rotation of a frame attachedto a rigid body, from F to F ′


O

z′ z

y

Mx

y ′

x ′


Rotation

Notations

m = (mx my mz)T and m′ = (m′x m′y m′z)T : coordinates ofpoint M respectively in F and F ′.

Then :

m = m′xx ′ + m′yy ′ + m′zz ′

=(x ′ y ′ z ′

)m′xm′ym′z

ConsequenceChange of basis equation (or coordinate transformation) :

m = Rm′


Rotation

Notations


Then :

m = m′xx ′ + m′yy ′ + m′zz ′

=(x ′ y ′ z ′

)m′xm′ym′z


m = Rm′


Rotation

Notations


Then :

m = m′xx ′ + m′yy ′ + m′zz ′

=(x ′ y ′ z ′

)m′xm′ym′z


m = Rm′


Rotation

Notations


Then :

m = m′xx ′ + m′yy ′ + m′zz ′

=(x ′ y ′ z ′

)m′xm′ym′z


m = Rm′


Rotation

O

z′ z

y

Mx

y ′

x ′

First interpretationChange of basis of the point coordinates

O

z′ z

y

Mx

y ′

x ′

Second interpretationRotation of S about O, characterized bymatrix R.Then m′ = initial coordinates of M in Fand m = final coordinates of M in F .


Rotation

Example

M

θ

y ′

z = z′O x

yx ′

M with initial coordinates (√

3 0 1)T .Give M coordinates after rotation R(z, θ).


Rotation

Solution

m =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

√301

=

√

3 cos θ√3 sin θ

1

.

For θ = π3 , it comes that m = (

√3

232 1)T .


Vector rotation

RemarkVector coordinates = difference between the coordinates of twopoints.

Then, rotation can be applied to v = m − n in F :

m − n = Rm′ − Rn′ = R(m′ − n′),

that is :v = Rv ′

with v ′ = m′ − n′


Rotation properties

NotationsIdentity matrix is denoted as I, whatever the dimension.

Orthogonality : RT R = I and det R = 1.Neutral element : identity matrix of dimension 3.Unique inverse : R−1 = RT .Combination of two successive rotations R1 and R2 :rotation R1R2.


Rotation properties


Orthogonality : RT R = I and det R = 1.

Neutral element : identity matrix of dimension 3.Unique inverse : R−1 = RT .Combination of two successive rotations R1 and R2 :rotation R1R2.


Rotation properties


Orthogonality : RT R = I and det R = 1.Neutral element : identity matrix of dimension 3.

Unique inverse : R−1 = RT .Combination of two successive rotations R1 and R2 :rotation R1R2.


Rotation properties


Orthogonality : RT R = I and det R = 1.Neutral element : identity matrix of dimension 3.Unique inverse : R−1 = RT .

Combination of two successive rotations R1 and R2 :rotation R1R2.


Rotation properties


Orthogonality : RT R = I and det R = 1.Neutral element : identity matrix of dimension 3.Unique inverse : R−1 = RT .Combination of two successive rotations R1 and R2 :rotation R1R2.


Combination of rotations

NotationsLet F ′ and F ′′ be two frames resulting from two successiverotations R1 and R2 of the reference frame F .

Non-commutativity of rotationsR1R2 6= R2R1.

Two possible cases :

second rotation with respect to the frame which resultsfrom the first rotation : (F ′′ results from the rotation of F ′about an axis attached to F ′)second rotation with respect to the same fixed frame as thefirst rotation (F ′′ results from the rotation of F ′ about anaxis attached to F)





Two possible cases :second rotation with respect to the frame which resultsfrom the first rotation : (F ′′ results from the rotation of F ′about an axis attached to F ′)

second rotation with respect to the same fixed frame as thefirst rotation (F ′′ results from the rotation of F ′ about anaxis attached to F)





Two possible cases :second rotation with respect to the frame which resultsfrom the first rotation : (F ′′ results from the rotation of F ′about an axis attached to F ′)second rotation with respect to the same fixed frame as thefirst rotation (F ′′ results from the rotation of F ′ about anaxis attached to F)


First case

Change of basis problemSecond rotation with respect to the frame which results fromthe first rotation.

NotationsM with coordinates m, m′, m′′ respectively in F , F ′ and F ′′

Combination : first caseAs m = R1m′ and m′ = R2m′′, then :

m = R1R2m′′.


First case

Change of basis problemSecond rotation with respect to the frame which results fromthe first rotation.


Combination : first caseCoordinates m of M in F = result of two successive rotationsapplied to a point with initial coordinates m′′


First case

Example

z ′′

O

π4

z z ′

M

x

x ′

x ′′π

y ′

m′′ = (√

2 0 0)T in F ′′ : coordinates of M in F ?


First case

Solution

m =

√

22 −

√2

2 0√2

2

√2

2 00 0 1

−1 0 0

0 1 00 0 −1

√200

=

−1−10

.

with the two following rotations :a first rotation of π4 about za second rotation of π about y ′


Second case

Successive rotationsProblem with successive rotations : the transformation of apoint with initial coordinates m′′ in F gives an intermediatepoint, which in turn gives a point with coordinates m in F after asecond rotation R2.


Combination : second caseConsequence :

m = R2(R1m′′)


Second case

Example

z ′′

O

π4

z z ′

x

x ′

π

Mx ′′

y

m′′ = (√

2 0 0)T in F ′′ : coordinates of M in F ?


Second case

Solution

m =

−1 0 00 1 00 0 −1

√

22 −

√2

2 0√2

2

√2

2 00 0 1

√2

00

=

−110

.

with the two following rotations :a first rotation of π4 about za second rotation of π about y


Orientation of a rigid body – direction cosines

NotationsRotation matrix of dimension 3× 3, to represent the rotationfrom frame F to frame F ′.

R =

xx yx zxxy yy zyxz yz zz

DefinitionElements of R=direction cosines, represent the coordinates ofthe three vectors of F ′ basis with respect to F .



RemarkThe columns in R are orthogonal to each other and so only twoof them are required :

R =

xx ∗ zxxy ∗ zyxz ∗ zz

.

ConsequenceDirection cosines computation limited to six parameters.



RemarkThese six parameters are related by three additional relations :

xxzx + xyzy + xzzz = 0x2

x + x2y + x2

z = 1

z2x + z2

y + z2z = 1

ConclusionMinimum set of three parameters : Euler angles ; roll, pitch, yawangles ; etc.


Orientation of a rigid body – Euler classical angles

DefinitionEuler classical angles = three successive rotations :

R(z, ψ), R(xψ, θ), R(zθ, ϕ)

with ψ, θ and ϕ : precession, nutation and spin.

x

z

ψ

xψ

zψ

θ

xθ

zθ

y yψ

yθϕ

zϕ

yϕ

xϕ



Every new rotation is performed with respect to a frame that haspreviously rotated :

R = R(z, ψ) R(xψ, θ) R(zθ, ϕ)

that is :

R =

0@cosψ − sinψ 0sinψ cosψ 0

0 0 1

1A0@1 0 00 cos θ − sin θ0 sin θ cos θ

1A0@cosϕ − sinϕ 0sinϕ cosϕ 0

0 0 1

1A=

0@cosψ cosϕ− sinψ cos θ sinϕ − cosψ sinϕ− sinψ cos θ cosϕ sinψ sin θsinψ cosϕ+ cosψ cos θ sinϕ − sinψ sinϕ+ cosψ cos θ cosϕ − cosψ sin θ

sin θ sinϕ sin θ cosϕ cos θ

1A



Inverse transformation = Euler angles from the direction co-sines :

if zz 6= ±1 :

ψ = atan2(zx ,−zy )θ = acos zzϕ = atan2(xz , yz)

if zz = ±1 :

θ = π(1− zz)/2ψ + zzϕ = atan2(−yx , xx )

and so ψ and ϕ are undetermined.


Orientation of a rigid body – roll, pitch, yaw angles

DefinitionRoll, pitch, yaw angles : three successive rotations :

R(x , γ), R(y , β), R(z, α)

with γ, β, and α roll, pitch and yaw angles.

x

z

y

βγ

α



Every new rotation is performed with respect to a fixed axis ofthe reference frame F :

R = R(z, α) R(y , β) R(x , γ)

that is :

R =

0@cosα − sinα 0sinα cosα 0

0 0 1

1A0@ cosβ 0 sinβ0 1 0

− sinβ 0 cosβ

1A0@1 0 00 cos γ − sin γ0 sin γ cos γ

1A=

0@cosα cosβ − sinα cos γ + cosα sinβ sin γ sinα sin γ + cosα sinβ cos γsinα cosβ cosα cos γ + sinα sinβ sin γ − cosα sin γ + sinα sinβ cos γ− sinβ cosβ sin γ cosβ cos γ

1A



Inverse transformation = roll, pitch, yaw angles derived form thedirection cosines :

if β 6= ±π2 :

α = atan2(xy , xx )

β = atan2(−xz ,√

x2x + x2

y )

γ = atan2(yz , zz)

if β = ±π2 :

α− sign(β) γ = atan2(zy , zx )or α− sign(β) γ = −atan2(yx , yy )

and so α and γ are undetermined.


Rigid-Body Transformation

DefinitionRigid-body transformation : (p, R) with p the translation of theorigin of the frame attached to the moving rigid body S and Rthe rotation of a frame attached to S.

O′

z′

y ′

O

z

xy

M

p

x ′


Rigid-Body Transformation

Notations

Let m = (mx my mz)T and m′ = (m′x m′y m′z)T be thecoordinates of a point M respectively in F and F ′.

Transformation equationRigid-body transformation : translation p of frame F and thenrotation R of the resulting frame to obtain F ′ :

m = p + Rm′


Homogeneous transformation matrix

DefinitionTo account for the rigid-body transformation in a linear way, thehomogeneous coordinates of point M are introduced :

m = (mx my mz 1)T = (mT 1)T .

(m1

)=

(R p0 1

)(m′

1

)

Consequence

m = T m′ with T =

(R p0 1

)Matrix T is the homogeneous transformation matrix.


Properties of rigid-body transformation

NotationsLet T , T1 and T2 be the rigid-body transformations (p, R),(p1, R1) and (p2, R2).

Combination : T1T2 =

(R1R2 R1p2 + p1

0 1

).

Neutral element : identity matrix of dimension 4.

Inverse : T−1 =

(RT −RT p0 1

).





(R1R2 R1p2 + p1

0 1

).


Inverse : T−1 =

(RT −RT p0 1

).





(R1R2 R1p2 + p1

0 1

).


Inverse : T−1 =

(RT −RT p0 1

).





(R1R2 R1p2 + p1

0 1

).


Inverse : T−1 =

(RT −RT p0 1

).


Rigid-body motion

DefinitionRigid-body transformation + time parameterization = rigid-bodymotion.

CharacterizationAngular velocity vector + linear velocity vector (of a chosenpoint)


Angular velocity vector

DefinitionAngular velocity vector Ω about the instantaneous rotation axisof S (counter-clockwise orientation).

O

z′ z

y

x

y ′

Ω

M

OM x ′vM


Velocity of a point of a rigid body

RotationLet Ω be the angular velocity vector S and vM be the velocity ofM ∈ S, with coordinates vM .

Linear velocity :

vM = Ω×OM,

or vM = Ω×m = Ω m,

with :

Ω =

0@ 0 −Ωz ΩyΩz 0 −Ωx−Ωy Ωx 0

1A

Linear velocity for combined rotation-translation

vM = p + Ω m.


Outline





Study framework

Only serial manipulators !We consider only the mechanical systems built from openpoly-articulated kinematic chains, called serial roboticmanipulators.


Description of open kinematic chains

DefinitionRobotic manipulator : n moving rigid bodies coupled by nrevolute or prismatic joints

L1 L2 Ln−1 Ln

joint

link link link link

joint joint jointjoint(link L0)ground

J1 J2 J3 Jn−1 Jn


Modified Denavit-Hartenberg parameters

Notationsi-th joint and i-th link : frame Fi = (Oi , x i , y

i, z i), with

i = 0, 1, . . . , n.

joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1


Frames F1 to Fn−1 placement

joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

Oi−1 is the point of Li−1 on the common normal to Li−1 and Li .

In the case of parallel axis, arbitrary choice of the perpendicularline.



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

x i−1 : unit vector of the common normal oriented from Li−1 to Li .

Arbitrary orientation when Li−1 and Li intersect or when Li−1 =Li (up, right or front).



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

z i−1 : unit vector of Li−1, with an arbitrary orientation (up, rightor front).



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

yi−1

: such that Fi−1 is a right-hand frame.


Frames F0 and Fn

ConventionFrame F0 : choose the simplest configuration.

Point On+1 : attached to the end effector.Frame Fn : such that On+1 ∈ (On, xn, zn).

O

z

x

y

On

On+1

zn

xn

an

rn+1


Frames F0 and Fn

ConventionFrame F0 : choose the simplest configuration.Point On+1 : attached to the end effector.

Frame Fn : such that On+1 ∈ (On, xn, zn).

O

z

x

y

On

On+1

zn

xn

an

rn+1


Frames F0 and Fn

ConventionFrame F0 : choose the simplest configuration.Point On+1 : attached to the end effector.Frame Fn : such that On+1 ∈ (On, xn, zn).

O

z

x

y

On

On+1

zn

xn

an

rn+1



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

αi−1 : angle between z i−1 and z i ,about x i−1.

ai−1 : distance on the commonnormal of axis Li−1 et Li , alongx i−1.θi : angle between x i−1 and x i ,about z i .ri : distance from the commonnormal to point Oi , along z i .



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

αi−1 : angle between z i−1 and z i ,about x i−1.ai−1 : distance on the commonnormal of axis Li−1 et Li , alongx i−1.

θi : angle between x i−1 and x i ,about z i .ri : distance from the commonnormal to point Oi , along z i .



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

αi−1 : angle between z i−1 and z i ,about x i−1.ai−1 : distance on the commonnormal of axis Li−1 et Li , alongx i−1.θi : angle between x i−1 and x i ,about z i .

ri : distance from the commonnormal to point Oi , along z i .



joint axisJi−1

x i−1

z i−1

ai−1

Oi−1

Ωi−1

Ji

joint axis

x iOi

x i

x i−1

z i

θi

ri

z i

z i

αi−1

αi−1 : angle between z i−1 and z i ,about x i−1.ai−1 : distance on the commonnormal of axis Li−1 et Li , alongx i−1.θi : angle between x i−1 and x i ,about z i .ri : distance from the commonnormal to point Oi , along z i .


ExampleSCARA-type manipulator. . .


Rigid-body transformation

Transformation parameterization :

Ti−1, i =

0BB@1 0 0 00 cosαi−1 − sinαi−1 00 sinαi−1 cosαi−1 00 0 0 1

1CCA| z

R(xi−1, αi−1)

0BB@1 0 0 ai−10 1 0 00 0 1 00 0 0 1

1CCA| z

translation ai−1xi−1

0BB@cos θi − sin θi 0 0sin θi cos θi 0 0

0 0 1 00 0 0 1

1CCA| z

R(zi , θi )

0BB@1 0 0 00 1 0 00 0 1 ri0 0 0 1

1CCA| z

translation ri zi

that is :

Ti−1, i =

cos θi − sin θi 0 ai−1

cosαi−1 sin θi cosαi−1 cos θi − sinαi−1 −ri sinαi−1sinαi−1 sin θi sinαi−1 cos θi cosαi−1 ri cosαi−1

0 0 0 1


Rigid-body transformation

Transformation parameterization :

Ti−1, i =

0BB@1 0 0 00 cosαi−1 − sinαi−1 00 sinαi−1 cosαi−1 00 0 0 1

1CCA| z

R(xi−1, αi−1)

0BB@1 0 0 ai−10 1 0 00 0 1 00 0 0 1

1CCA| z

translation ai−1xi−1

0BB@cos θi − sin θi 0 0sin θi cos θi 0 0

0 0 1 00 0 0 1

1CCA| z

R(zi , θi )

0BB@1 0 0 00 1 0 00 0 1 ri0 0 0 1

1CCA| z

translation ri zi

which writes :

Ti−1, i =

(Ri−1, i pi−1, i

0 1

)with Ri−1, i the rotation between frames Fi−1 and Fi and pi−1, ithe translation between the same frames.


Prismatic joint

joint axisJi

Oi

On

qi z i

pi

= qi z i

Velocity of On and angular velocity of Fn :

pi

= qi z i ,

Ωi = 0.


Revolute joint

joint axisJi

Oi

On

pi,n

Ωi = qi z i

pi

= qiz i × pi,n

Velocity of On and angular velocity of Fn :

pi

= qi z i × pi,n,

Ωi = qi z i .


Differential kinematic relations, general case

NotationsLink type identified by parameter σi :

σi =

0, for a revolute joint,1, for a prismatic joint.

Velocity of the end effector frame in On

pi

= (σi z i + σi z i × pi,n

)qi ,

Ωi = (σi z i) qi .


Outline





Configuration

DefinitionConfiguration of a mechanical system : set of minimal numberof parameters, that give the position of any point of the systemin a given frame.

Robotic manipulators caseConfiguration of a robotic manipulator : vector q of nindependent coordinates called the generalized coordinates.The set of admissible generalized coordinates is theconfiguration space N .

Generalized coordinates : rotation angles for revolute joints,translation values for prismatic joints.


Pose

DefinitionPose of a rigid body : position and orientation of this rigid bodyin a given frame.

Robotic manipulators casePose of the end effector of a robotic manipulator : vector x of mindependent operational coordinates. The set of admissibleoperational coordinates is theoperational spaceM, ofdimension m 6 6.

Depends on the task (planar cases, positioning only . . .) and onthe parameterization of orientations.


Direct kinematics

DefinitionDirect kinematic model (DKM) of a robotic manipulator : pose ofthe end effector as a function of the robot configuration :

f : N −→ Mq 7−→ x = f (q).

General case

Expression of x = (x1 x2 x3 x4 x5 x6)T , with (x1 x2 x3)T positioncoordinates in F0 and (x4 x5 x6)T orientation coordinates, as afunction of q = (q1 q2 . . . qn)T .

Often using partial direction cosines.


DKM computation

Orientation computed from the rotation matrix between theground frame and the end effector frame.

Position (x1 x2 x3)T of point On+1 computed from theposition (px py pz)T of point On in F0, given thecoordinates (an 0 rn+1)T of On+1 in Fn :

x1 = px + anxx + rn+1zx

x2 = py + anxy + rn+1zy

x3 = pz + anxz + rn+1zz


DKM computation

Orientation computed from the rotation matrix between theground frame and the end effector frame.Position (x1 x2 x3)T of point On+1 computed from theposition (px py pz)T of point On in F0, given thecoordinates (an 0 rn+1)T of On+1 in Fn :

x1 = px + anxx + rn+1zx

x2 = py + anxy + rn+1zy

x3 = pz + anxz + rn+1zz


Practical considerations

Computation of On coordinates and of the direction cosines :

T0,n(q) = T0,1(q1) T1,2(q2) . . . Tn−1,n(qn).

Rules

For i , j , . . . in 1,2, . . . ,n, write (for the sake of simplicity) :

Si = sin qi

Ci = cos qi

Si+j = sin (qi + qj)

Ci+j = cos (qi + qj)




T0,n(q) = T0,1(q1) T1,2(q2) . . . Tn−1,n(qn).

Rules

Each new mathematical operation (addition, multiplication) : in-troduction of a new intermediate variable.




T0,n(q) = T0,1(q1) T1,2(q2) . . . Tn−1,n(qn).

Rules

Reverse product computation : no computation of the secondcolumn of the rotation matrix.




T0,n(q) = T0,1(q1) T1,2(q2) . . . Tn−1,n(qn).

Rules

First compose transformations with particular properties, in par-ticular rotations with parallel axis.




Inverse kinematics

DefinitionInverse kinematic model (IKM) : the (one or several)configurations that correspond to a given pose of the robot endeffector :

f−1 : M −→ Nx 7−→ q = f−1(x).

SolvabilityExistence of solutions.

If n < m : no solution.If n = m : finite number of solutions (in general).If n > m : infinite number of solutions.


Computation

IKM computationNo systematic analytic method.

Set of nonlinear equations formed by the DKM to be solved.When n = 6, the system is generally equipped with a sphericalwrist that allows some decoupling between orientation andposition, which helps solving the IKM :

px = x1 − anxx − rn+1zx ,

py = x2 − anxy − rn+1zy ,

pz = x3 − anxz − rn+1zz .

Unfortunately the next steps require much more complexcomputations to solve for the qi , i = 1, 2, . . . , n as functions ofthe obtained px , py , pz and of the direction cosines.




Differential kinematics

DefinitionDirect differential kinematic model (DDKM) : relation betweenthe operational velocities x and the generalized velocities q :

x = Jq

where J = J(q) is the Jacobian matrix of function f , ofdimension m × n :

J : TqN −→ TxM

q 7−→ x = Jq, with J =∂f∂q.



DDKM computationDifferentiation of the DKM for simple structures or . . .

First : velocity of On and angular velocity of frame Fn

p =n∑

i=1

(σi z i + σi z i × pi,n

)qi ,

Ω =n∑

i=1

(σi z i) qi .





In a vector representation :(pΩ

)= Jg q

Jg =

„σ1z1 + σ1z1 × p

1,nσ2z2 + σ2z2 × p

2,n. . . σnzn + σnzn × p

n,nσ1z1 σ2z2 . . . σnzn

«





In F0 : (pΩ

)= Jg q




Second : computation of the velocity of point On+1 and ofthe derivatives of the orientation parameters of frame Fn

Position of On+1 :x1x2x3

=

pxpypz

+ an

xxxyxz

+ rn+1

zxzyzz





Velocity of On+1 :x1x2x3

=

pxpypz

+

ΩxΩyΩz

×an

xxxyxz

+ rn+1

zxzyzz




Second : computation of the velocity of point On+1 and ofthe derivatives of the orientation parameters of frame Fnx1

x2x3

=

pxpypz

+ D

ΩxΩyΩz

with D =

0 anxz + rn+1zz −anxy − rn+1zy−anxz − rn+1zz 0 anxx + rn+1zxanxy + rn+1zy −anxx − rn+1zx 0

.





Derivatives of the orientation parameters of frame Fn :x4x5. . .xm

= C

ΩxΩyΩz



Finally :DDKM :

x =

(I D0 C

)(pΩ

)=

(I D0 C

)Jg q,

Jacobian matrix :

J =

(I D0 C

)Jg .


Practical rules

For analytical calculations, it is worth using :

Jg =

„R0,1 0

0 R0,1

«„R1,2 0

0 R1,2

«. . .

„Rk−1,k 0

0 Rk−1,k

«„I −pk+1,n|Fk0 I

«Jk+1|Fk

with :

k = Int(n2 )

robot with a spherical wrist : p4,6 = 0pk+1,n|Fk : antisymmetric matrix associated to theprojection of pk+1,n in Fk

Jk+1|Fk : projection in Fk

Jk+1 =

σ1z1 + σ1z1 × p

1,k+1σ2z2 + σ2z2 × p

2,k+1. . . σnzn + σnzn × p

n,k+1σ1z1 σ2z2 . . . σnzn

!


Practical rules


Jg =

„R0,1 0

0 R0,1

«„R1,2 0

0 R1,2

«. . .

„Rk−1,k 0

0 Rk−1,k


«Jk+1|Fk

with :k = Int(n

2 )



Jk+1 =

σ1z1 + σ1z1 × p

1,k+1σ2z2 + σ2z2 × p

2,k+1. . . σnzn + σnzn × p


!


Practical rules


Jg =

„R0,1 0

0 R0,1

«„R1,2 0

0 R1,2

«. . .

„Rk−1,k 0

0 Rk−1,k


«Jk+1|Fk

with :k = Int(n

2 )

robot with a spherical wrist : p4,6 = 0

pk+1,n|Fk : antisymmetric matrix associated to theprojection of pk+1,n in Fk


Jk+1 =

σ1z1 + σ1z1 × p

1,k+1σ2z2 + σ2z2 × p

2,k+1. . . σnzn + σnzn × p


!


Practical rules


Jg =

„R0,1 0

0 R0,1

«„R1,2 0

0 R1,2

«. . .

„Rk−1,k 0

0 Rk−1,k


«Jk+1|Fk

with :k = Int(n

2 )



Jk+1 =

σ1z1 + σ1z1 × p

1,k+1σ2z2 + σ2z2 × p

2,k+1. . . σnzn + σnzn × p


!


Practical rules


Jg =

„R0,1 0

0 R0,1

«„R1,2 0

0 R1,2

«. . .

„Rk−1,k 0

0 Rk−1,k


«Jk+1|Fk

with :k = Int(n

2 )



Jk+1 =

σ1z1 + σ1z1 × p

1,k+1σ2z2 + σ2z2 × p

2,k+1. . . σnzn + σnzn × p


!




The End

Further questions ?

Documents

Introduction to Robotics - Unistraeavr.u-strasbg.fr/~bernard/education/houston/slides_houston... · Introduction to Robotics Bernard BAYLE Joint course University of Strasbourg, University