Kinematics - cmp.felk.cvut.czcmp.felk.cvut.cz/.../teaching/robotics/lectures/kinematika-note.pdf · 2/31 Kinematics ± Terminology J5 axis J4 axis Fore arm Elbow bloc k J3 axis Upper

1/31Kinematics

Kinematics analyzes the geometry of a manipulator, robot ormachine motion. The essential concept is a position.

Statics deals with the forces and moments which are aplied on themechanism at rest. The essential concept is a stiffness.

Dynamics analyzes the forces and moments which result frommotion and acceleration of a mechanism and a load.

The terms and laws studied can be applied to robot-industrialmanipulator as well as to any other machine with movingcomponents. We will refer here to robot and will use some terms usedin robotics (like end effector) but any machine could and shall bestudied in similar way when position, stiffness or dynamics of thesystem is important.

ROBOTICS, Vladimır Smutny Slide 1, Page 1

2/31

Kinematics – Terminology

J5 axisJ4 axis

Fore armElbow block

J3 axis

Upper arm

J2 axis

Base

J1 axis

Shoulder

J6 axis

Link is the rigid part of the robot body (e.g.forearm).

Joint is a part of the robot body whichallows controlled or free relative motionof two links (connection element).

End effector is the link of the manipulatorwhich is used to hold the tools (gripper,spray gun, welding gun...).

Base is the link of the manipulator which isusually connected to the ground and isdirectly connected to the world coordi-nate system.

Kinematic pair is a pair of links which rela-tive motion is bounded by the joint con-necting them (e.g. base and shoulderconnected by J1 axis).

Joint can be either controlled or freely moving. For con-trolled joint there is an actuator mounted in it and control

system can change its position. Freely moving joint changesits status according position of other joints.


3/31

Kinematics – Terminology II

Kinematic chain is a set of links connected by joints. Kinematic chain can berepresented by a graph. The vertices represent links and edges represent joints.

Mechanism is a kinematic chain when one of its links is fixed to the ground.

Open kinematic chain Mixed kinematic chain Parallel manipulatorsis the chain which can be de-scribed by acyclic graph. contains in its graph a loop. consist of equivalent loops.


3/31

Kinematics – Degrees of Freedom

Degrees of freedom (less formal definition) is a number ofindependent parameters needed to specify the position of themechanism completely. Examples:

Rigid body in a 2D space e.g. plane has 3 DOF.

Rigid body in a 3D space has 6 DOF.

A point in a plane has 2 DOF.

A point in a 3D space has 3 DOF.


3/31


Degrees of freedom (less formal definition) is a number ofindependent parameters needed to specify the position of themechanism completely. Examples:

Rigid body in a 2D space e.g. plane has 3 DOF.

Rigid body in a 3D space has 6 DOF.

A point in a plane has 2 DOF.

A point in a 3D space has 3 DOF.


4/31


The DOF is important notion not only in robotics. Few more definitions are related to it:

Ambient space - the space robot/mechanism lives in, usually E2 (the plane - planarmanipulator) or E3 (space). It is Euclidean space.

Operational space.

is the subspace of the ambient space occupied by any of therobot part during any of possible robot motions. The operatio-nal (3D, physical) space shall be guarded by fence, doors orinvisible bariers to prevent injuries of both robot and humans.

Work envelope (working space).

is the subspace of the ambient space wherethe robot can reach by the end effector. Thecuts through this space are usually drawn inthe technical specifications by manufactu-rers. The work envelope has actually littleuse in the practice, it can just provide basicnotion where the robot can work.


5/31


We usually need to study the position of the end effector or the tool fixed to it. Letus assume the end effector or tool is a rigid body.

The rigid body position in the 3D ambient space can be described by sixparameters. The semantics and values depend on the chosen parametrization,e.g. position of the reference point on it (3 parameters) and 3 angles.

The “position” space is the 6D (3D for planar case) space representing allpossible positions of rigid body in 3D (2D) ambient space.

The end effector position can be studied in this 6D space position space.

The working space is a subspace of the position space containing positionswhich can be reached by end effector (tool). All required end effector positionsshall of course lie in the working space, so the feasibility of the particular robotuse (its reach) shall be studied in this space.

The joint space a space of all possible??????????????????

joints space trajectories

The DOF is a dimension of the subspace which contains ??????????????

Revolute joint is preferred in machinery as it can be eas-ily and cheply manufactured, it has low friction, good rigid-

ity. We will study mainly robots with revolute and prismaticjoints.


6/31

Types of kinematic pairs

Symbol Name has/removes DOF

Spherical 3 / 3

Revolute 1 / 5

Prismatic 1 / 5

Cylindrical 2 / 4

Flat 3 / 3


7/31

Typical structure of manipulators

Cartezian Cylindrical Spherical

Angular robot has large working space compare to dimen-sions. Scara is particularly suited for operations above hori-

zontal plane, where three vertical axes does not work againstgravity.


8/31

Typical structure of manipulators II

Angular SCARA

Animations taken from Masud Salimian’s web page

Body in the coordinate system

The rigid body in a plane has 3 degree of freedom.The same body in 3D space has 6 degree of freedom:Questions:How many DOF has a rubber tape?

Let the coordinate system of a base is O−xyz. The coor-dinate system of a body is O′ − xbybzb. The description of a

coordinate system O′−xbybzb in coordinate system of a baseis:

~OO′ = xo =

xo

yo

zo

(n, t, b) .

Let us form a matrix R = (n, t,b), n, t, b are unit andorthogonal vectors, then a matrix R is ortonormal, that isR−1 = RT .


10/31

Body in the coordinate system

Point in 3D - described by three coordinates.

Rigid body in 3D - described by 6 coordinates:

� 3 coordinates of reference point x0 =æçççççççè

x0y0z0

ö÷÷÷÷÷÷÷ø

� orientation could be described e.g. by:

ê coordinates of vectors attached to the body (n, t,b),

ê Euler angles (Φ, Θ,Ψ),

ê rotational matrix R.

Euler angles

The matrix R has nine coeficients, but its dimension is onlythree, the bounding condition is unit size and orthogonalityof vectors n, t, b:

nT t = 0 tT b = 0 bT n = 0|n| = 1 |t| = 1 |b| = 1

The matrix R can be contructed easily using Euler angles,see Fig. ??:

1. Rotate the coordinate system O−xyz around the axisz by the angle φ. We will get O − x′y′z.

2. Rotate the coordinate system O−x′y′z around the axisx′ by the angle θ. We will get O − x′y′′z′′.

3. Rotate the coordinate system O − x′y′′z′′ around theaxis z′′ by the angle ψ. We will get O − xbybzb.

R = Rz(φ)Rx′(θ)Rz′′(ψ)

Rz(φ) =

cosφ − sinφ 0sinφ cosφ 0

0 0 1

(1)

Rx′(θ) =

1 0 00 cos θ − sin θ0 sin θ cos θ

(2)

Rz′′(ψ) =

cosψ − sinψ 0sinψ cosψ 0

0 0 1

(3)

R =

cosφ cosψ − cos θ sinφ sinψ − cos θ cosψ sinφ− cosφ sinψ sinφ sin θcosψ sinφ+ cosφ cos θ sinψ cosφ cos θ cosψ − sinφ sinψ − cosφ sin θ

sin θ sinψ cosψ sin θ cos θ

(4)

Euler angles define uniquely the rotation, the same ro-tation can be reached by different triples of angles. Otherdefinitions like roll–pitch–yaw define similar angles with sim-ilar properties but with different equations. If the matrix Ris given, Euler angles can be calculated by the comparison ofthe matrix coeficients r33, r32, r23.


12/31

Rotation Matrix Resulting from Euler Anglesæçççççççç

è

Cos(Φ) Cos(Ψ) - Cos(Θ) Sin(Φ) Sin(Ψ) -Cos(Θ) Cos(Ψ) Sin(Φ) - Cos(Φ) Sin(Ψ) Sin(Φ) Sin(Θ)Cos(Ψ) Sin(Φ) + Cos(Φ) Cos(Θ) Sin(Ψ) Cos(Φ) Cos(Θ) Cos(Ψ) - Sin(Φ) Sin(Ψ) -Cos(Φ) Sin(Θ)

Sin(Θ) Sin(Ψ) Cos(Ψ) Sin(Θ) Cos(Θ)

ö÷÷÷÷÷÷÷÷

ø

We know the coordinates of a point in coordinate system

O′ − xbybzb: xb =

uvw

and we are looking for its coordi-

nates in coordinate system O − xyz: x =

xyz

.

~OP ′ = ~OO′ + ~O′A+ ~AB + ~BP

x = xo + un + vt + wb

x = xo + Rxb

Inverse transform:

xb = −RT xo + RT x


13/31Coordinate transformation

Homogeneous coordinates

Let us define homogenneous coordinates:Euclidean -metric Homogeneous - projective

x =

xyz

⇒ x =

xyz1

x =

x/wy/wz/w

⇐ x =

xyzw

∧ w 6= 0

does not exist(point at infinity) ⇐ x =

xyz0

It is possible to show, that

x = Axb,

where A is a matrix 4x4:

A =[

R xo

0 0 0 1

]Inverse matrix:

A−1 =[

RT −RT xo

0 0 0 1

]Consecutive coordinate transformations, see Fig. ??:

x0 = A01A

12A

23A

34 . . .A

n−1n xn. (5)


14/31Consecutive coordinate transformations

Open kinematic chain modelling

Open kinematic chain is formed by the sequence of links con-nected by joints. If we know the description joints using ge-

ometrical tranformations we can easily find transformationof point coordinates from end effector coordinate system tobase coordinate system and vice versa. This transformationis called kinematic equations.


15/31

Open kinematic chain

Unique and efficient description of transformations can befound by the method Denavit-Hartenberg. The descriptionis then called Denavit-Hartenberg notation. See fig. ??. Wedescribe the joint i.

1. Find the axes of rotation of joints i− 1, i, i+ 1.

2. Find the common normal of joint axes i − 1 and i andaxes of joints i a i+ 1.

3. Find points Oi−1, Hi, Oi.

4. Axis zi shall be placed into the axis of the joint i+ 1.

5. Axis xi shall be placed into the common normal HiOi.

6. Axis yi forms with the other axes right-hand coordinatesystem.

7. Name the distance of points |Oi−1, Hi| = di.

8. Name the distance of points |Hi, Oi| = ai.

9. Name the angle between common normals θi.

10. Name the angle between axes i, i+ 1 αi.

11. The origin of a base coordinate system Oo can be placedanywhere on the joint axis and axis x0 can be orientedarbitrarily. For example to get d1 = 0.

12. The origin On of the end effector coordinate system andorientation of the axis zn can be placed arbitrarily whenother rules hold.

13. When the axis of two consecutive joints are parallel, thecommon normal position can be placed arbitrarily, e.g.to get di = 0.

14. The position of joint axis can be arbitrarily chosen forprismatic joints.


19/31Base and end effector coordinate frames in DH

The tranformation in the joint is uniquelly described byfour parameters ai, di, αi, θi. Parameters ai, αi are constant,one of the parameters di, θi is changing when the joint moves.

The joints are usually:

• Revolute, then di is constant and θi is changing,

• Prismatic, then θi is constant and di is changing.

The matrix of transformation A can be calculated

Ai−1i = Ai−1

int Ainti ,

where

Ai−1int =

cos θi − sin θi 0 0sin θi cos θi 0 0

0 0 1 di

0 0 0 1

,

Ainti =

1 0 0 ai

0 cosαi − sinαi 00 sinαi cosαi 00 0 0 1

.

It can be shown, that:

Ai−1i =

cos θi − sin θi cos αi sin θi sin αi ai cos θi

sin θi cos θi cos αi − cos θi sin αi ai sin θi

0 sin αi cos αi di

0 0 0 1

.

Denote qi the parameter θi, di, which is changing. Ex-pression 5 can be redrawn to

x0 = A01(q1)A

12(q2)A

23(q3)A

34(q4) . . .A

n−1n (qn)xn.

For each value of the vector q = (q1, q2, q3, q4, . . . qn) ∈ Q =Rn we can calculate coordinates of point P in base coordinatesystem from given P coordinates in end effector coordinatesystem and vice versa.

Kinematic equation are always solvable analytically.


20/31

5-R-1-P manipulator

Inverse kinematics

We call inverse kinematics the task when there is give a ma-trix

T(q) = A01(q1)A

12(q2)A

23(q3)A

34(q4) . . .A

n−1n (qn). (6)

and we are looking for values of vector q. The system ofnonlinear equations (usually trigonometric) is typically notpossible to solve analytically.

Solving inverse kinematics:

• Analytically, if possible. There is not a unique descrip-tion, how to solve the system.

• Numerically.

• Look up table, precalculated for the working spaceW ⊂ Q.

There are a manipulator structures, which can be solved an-alytically. We call them solvable.

The sufficient condition of solvability is e.g. when the 6DOF robot has three consecutive revolute joint with axes in-tersecting in one point.

The other property of inverse kinematics is ambiquity ofsolution in singular points. There is often the subspace Qs ofthe space Q, which gives the same T.

∀q ∈ Qs : T(q) = T

To decide which q solving 6 to select has to be taken intoaccount mainly:

1. Is the selected value q applicable, i.e. can the robotbe sent to q?

2. How to reach the singular point. The function q shallallways be a continuous function of time. The preceed-ing values of q should make with the selected value con-tinous function of time.

3. How to continue from the singular point? The futurevalues of q should form with the selected value conti-nous function of time.

4. Will not the selected value of q guide us to the situationwhere we will not be able to satisfy above conditions?

5. Will the required operational space limit us duringmanipulation? The example is the insertion of the seatinto car.

We design sometimes redundant robots (with more de-grees of freedom, e.g. 8), to increase the space Qs, fromwhich we select q to allow more freedom to fulfill above re-quirements.

Think about following:

• Is it possible to design the robot with prismatic jointsonly which can position arbitrarily the rigid body in 3Dspace? Why?

• Choose some manipulating task and design the struc-ture of redundant robot for it.


Documents

Kinematics - cmp.felk.cvut.czcmp.felk.cvut.cz/.../teaching/robotics/lectures/kinematika-note.pdf · 2/31 Kinematics ± Terminology J5 axis J4 axis Fore arm Elbow bloc k J3 axis Upper