Kinematics - Introduction. Robotics. Kinematics - Introduction. Vladimír Smutný
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1 Kinematics - Introduction Robotics Kinematics - Introduction Vladimír Smutný Center for Machine Perception Czech Institute for Informatics, Robotics, and Cybernetics (CIIRC) Czech Technical University in Prague ROBOTICS: Vladimír Smutný Slide 1, Page 1
2 Mechanics Kinematics analyzes the geometry of a manipulator, robot or machine motion. The essential concept is a position. Statics deals with the forces and moments which are aplied on the mechanism at rest. The essential concept is a stiffness. Dynamics analyzes the forces and moments which result from motion and acceleration of a mechanism and a load. The terms and laws studied can be applied to robot-industrial manipulator as well as to any other machine with moving components. We will refer here to robot and will use some terms used in robotics (like end effector) but any machine could and shall be studied in similar way when position, stiffness or dynamics of the system is important ROBOTICS: Vladimír Smutný Slide 2, Page 2
3 Kinematics Terminology 3/34 J5 axis J4 axis Fore arm Elbow block Link is the rigid part of the robot body. Joint is a part of the robot body which allows controlled or free relative motion of two links (connection element). J6 axis J3 axis Upper arm End effector is the link of the manipulator which is used to hold the tools (gripper, spray gun, welding gun...). Shoulder J1 axis J2 axis Base is the link of the manipulator which is usually connected to the ground and is directly connected to the world coordinate system. Base Kinematic pair is a pair of links which relative motion is bounded by the joint connecting them. Joint can be either controlled or freely moving. For controlled joint there is an actuator mounted in it and control system can change its position. Freely moving joint changes its status according position of other joints. ROBOTICS: Vladimír Smutný Slide 3, Page 3
4 Kinematics Terminology II Kinematic chain is a set of links connected by joints. Kinematic chain can be represented 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. 4/34 ROBOTICS: Vladimír Smutný Slide 4, Page 4
5 Kinematics Terminology III Open kinematic chain Hybrid kinematic chain Parallel manipulators is the chain which can be described consist of equivalent loops. by acyclic graph. contains in its graph a loop. 5/34 ROBOTICS: Vladimír Smutný Slide 5, Page 5
6 Robot Graph Open Kinematic Chain 6 5 F 4 11 G 10 K 7 E L 12 M 13 6/34 H N 7 13 G M 6 12 F L 5 11 E K 4 10 N H J D 9 3 I C 8 2 D J 3 B 9 C 1 I 2 A 8 B 1 A ROBOTICS: Vladimı r Smutny Slide 6, Page 6
7 Robot Graph Hybrid Kinematic Chain 7/34 J 9 F 5 G H I 12 I G H J E 9 C 10 D F B C D 2 3 E 4 1 B A 1 A ROBOTICS: Vladimı r Smutny Slide 7, Page 7
8 Robot Graph Parallel Stewart/Gough Platform 8/34 G G F E F E D D C B A C ROBOTICS: Vladimı r Smutny B A Slide 8, Page 8
9 Kinematics Degrees of Freedom Degrees of freedom (less formal definition) is a number of independent parameters needed to specify the position of the mechanism completely. Examples: A point in a plane has 2 DOF. A point in a 3D space has 3 DOF. Rigid body in a 2D space e.g. plane has 3 DOF. Rigid body in a 3D space has 6 DOF ROBOTICS: Vladimír Smutný Slide 9, Page 9
10 Flange downward limit line(dotted line) Restriction on wide angle in the front section Note5) Restriction on wide angle in the front section Note4) Restriction on wide angle in the rear section Note1) J2+J degree when -45 degree J2 < 15 degree. Note2) J2+J3 8 degree when J1 75 degree, J2 < -45 degree. Note3) J2+J3-40 degree when J1 > 75 degree, J2 < -45 degree. Restriction on wide angle in the front section Note4) J3-40 degree when -105 degree J1 95 degree, J2 123 degree. Note5) J2 110 degree when J1 < -105 degree, J1 < -95 degree. However, J2 - J3 150 degree when 85 degree J2 110 degree Restriction on wide angle in the rear section Note2) Restriction on wide angle in the rear section Note3) Restriction on wide angle in the rear section Note1) P-point path: Reverse range (alternate long and short dash line) P-point path: Entire range (solid line) Areas as restricted by Note1) and Note3) within the operating range Kinematics Degrees of Freedom 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 E 2 (the plane - planar manipulator) or E 3 (space). It is Euclidean space. Operational space. 10/34 2 Robot arm is the subspace of the ambient space occupied by any of the robot part during any of possible robot motions. Work envelope (working space). R526 R R202 R258 is the subspace of the ambient space where the robot can reach by the end effector P R611 R173 Fig.2-5 : Operating range diagram : RV-6S/6SC 2-14 Outside dimensions Operating range diagram 17 R R R287 R331 R The operational (3D, physical) space shall be guarded by fence, doors or invisible bariers to prevent injuries of both robot and humans. The cuts through the work envelope are usually drawn in the technical specifications by manufacturers. The work envelope has actually little use in the practice, it can just provide basic notion where the robot can work. ROBOTICS: Vladimír Smutný Slide 10, Page 10
11 R Work envelope - example Robot arm 11/34 Flange downward limit line(dotted line) P R280 Restriction on wide angle in the rear section Note2) P-point path: Reverse range (alternate long and short dash line) P-point path: Entire range (solid line) Restriction on wide angle in the front section Note5) R287 R280 Restriction on wide angle in the rear section Note3) 100 Restriction on wide angle in the rear section Note1) R R331 R R526 R R331 Areas as restricted by Note1) and Note3) within the operating range R258 Restriction on wide angle in the front section Note4) 76 R Restriction on wide angle in the rear section Note1) J2+J degree when -45 degree J2 < 15 degree. Note2) J2+J3 8 degree when J1 75 degree, J2 < -45 degree. Note3) J2+J3-40 degree when J1 > 75 degree, J2 < -45 degree. Restriction on wide angle in the front section Note4) J3-40 degree when -105 degree J1 95 degree, J2 123 degree. Note5) J2 110 degree when J1 < -105 degree, J1 < -95 degree. However, J2 - J3 150 degree when 85 degree J2 110 degree Fig.2-5 : Operating range diagram : RV-6S/6SC 2-14 Outside dimensions Operating range diagram Flange downward limit line(dotted line) P R280 Restriction on wide angle in the rear section Note2) Restriction on wide angle in the front section Note5) R287 R280 Restriction on wide angle in the rear section Note3) 100 Restriction on wide angle in the rear section Note1) R331 R173 R R331 Areas as restricted by Note1) and Note3) within the operating range Restriction on wide angle in the front section Note4) Fig.2-5 : Operating range diagram : RV-6S/6SC 2-14 Outside dimensions Operating range diagram Restriction on wide angle in the rear section Note1) J2+J degree when -45 degree J2 < 15 degree. Note2) J2+J3 8 degree when J1 75 degree, J2 < -45 degree. Note3) J2+J3-40 degree when J1 > 75 degree, J2 < -45 degree. Restriction on wide angle in the front section Note4) J3-40 degree when -105 degree J1 95 degree, J2 123 degree. Note5) J2 110 degree when J1 < -105 degree, J1 < -95 degree. However, J2 - J3 150 degree when 85 degree J2 110 degree. ROBOTICS: Vladimír Smutný Slide 11, Page 11
12 Kinematics Degrees of Freedom 12/34 We usually need to study the position of the end effector or the tool fixed to it. Let us assume the end effector or tool is a rigid body. The rigid body position in the 3D ambient space can be described by six parameters. 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 all possible 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 positions which can be reached by end effector (tool). All required end effector positions shall of course lie in the working space, so the feasibility of the particular robot use (its reach) shall be studied in this space. ROBOTICS: Vladimír Smutný Slide 12, Page 12
13 Types of kinematic pairs Symbol Name has (f)/constraints (c) DOF Revolute 1 / 5 Prismatic 1 / 5 Cylindrical 2 / 4 Spherical 3 / 3 Flat 3 / Revolute joint is preferred in machinery as it can be easily and cheply manufactured, it has low friction, good rigidity. We will study mainly robots with revolute and prismatic joints. ROBOTICS: Vladimír Smutný Slide 13, Page 13
14 Degrees of Freedom Grübler (Kutzbach) criterion The number of degrees of freedom for mechanism: c i number of constraints imposed by joint i, f i number of DOF permitted by joint i, n number of links in the mechanism (mechanism has one link fixed), j number of joints in the mechanism (all shall be binary), λ number of DOF of the single rigid body in ambient space, F number of DOF of the whole mechanism. j F = λ(n j 1) + f i, alternatively F = λ(n 1) i=1 j c i. i= As a homework think about telescopic radio antena with 5 cylinders. How many degrees of freedom has a final cylinder? Does it hold equality in Grübler (Kutzbach) criterion? Where is the singularity? ROBOTICS: Vladimír Smutný Slide 14, Page 14
15 Grübler (Kutzbach) criterion, Nr. of DOF Example c i number of constraints imposed by joint i, 15/34 f i number of DOF permitted by joint i, n number of links in the mechanism (mechanism has one link fixed), j number of joints in the mechanism (all shall be binary), λ number of DOF of the single rigid body in ambient space, F number of DOF of the whole mechanism. alternatively j F = λ(n j 1) + f i, i=1 j F = λ(n 1) c i. i=1 Consider a transom mechanism on the figure. It models a window opened by sliding handle. The system could be viewed as planar or spatial. The number of degrees removed by the joint will be denoted by planar/spatial numbers. The number of links n = 4. The links are: 1 window 2 connecting bar 3 sliding handle 4 window frame They are connected by joints: A window hinge, revolute, removing DOF 2/5, B window bar hinge, revolute, 2/5, C bar slider joint, revolute, 2/5, D 1st handle holder, prismatic, 2/5, E 2nd handle holder, prismatic, 2/5. Let us insert the data into Gruebler formula: Planar case: F = 3(4 1) ( ) = 1 This number is apparently wrong. Is it? Actually it is not, as the formula does not consider the colinearity of E and D joint which shall model just one prismatic joint. The corrected planar case: F = 3(4 1) ( ) = 1 This gives the correct result of 1 DOF of the mechanism. Spatial case (considering colinearity of D and E): F = 6(4 1) ( ) = 2 Again unexpected result, but where is the mistake this time? Think about it before reading the solution. The problem is that correct function of transom mechanism assumes parallelity of A, B, and C joint axes. We should model (and manufacture) them as having more freedom. Case D, E being cylindrical and colinear, B spherical: F = 6(4 1) ( ) = 1 Case B, C being spherical: F = 6(4 1) ( ) = 2 This means the link bar 2 could freely rotate around axis passing through B and C. Case A cylindrical, B spherical: F = 6(4 1) ( ) = 1 Another practical solution is to make B and C as well as D and E slightly flexible, so minor nonparallelity of joints arrangement could be absorbed. It is cheap, but the mechanism is not nicely sliding. Another, rather expensive, solution is to make individual parts adjustable, so during the instalation, the best position is found and fixed. ROBOTICS: Vladimír Smutný Slide 15, Page 15
16 DOF Parallel Manipulator Example 16/34 ROBOTICS: Vladimír Smutný Slide 16, Page 16
17 Parallel Manipulator - Scheme and Dimensions 17/34 y l5 l4 l2 l3 l6 d2 d3 l7 d4 l10 l1 d1 l8 l9 l11 L x ROBOTICS: Vladimír Smutný Slide 17, Page 17
18 Parallel Manipulator Links and Joints 18/34 y B l2 2 C 3 l3 D l4 N d2 E 7 l5 4 F l6 9 G 1 l1 M d1 6 O 8 d3 H 10 l8 l7 12 K P d4 l10 11 L l11 l9 I J 13 L 5 A x ROBOTICS: Vladimír Smutný Slide 18, Page 18
19 Parallel Manipulator Systematic Angles 19/34 y β4 β5 l2 β3 l3 l4 l5 l6 β2 d2 β8 d3 l7 d4 l10 l1 d1 β6 l8 β7 l9 l11 L β1 x ROBOTICS: Vladimír Smutný Slide 19, Page 19
20 Parallel Manipulator Dimensions Orientation 20/34 y β4 β5 l2 β3 l3 l4 l5 l6 β2 d2 β8 d3 l7 d4 l10 l1 d1 β6 l8 β7 l9 l11 L β1 x ROBOTICS: Vladimír Smutný Slide 20, Page 20
21 Parallel Manipulator Loops 21/34 y β4 β5 l2 β3 l3 l4 l5 2 l6 β2 d2 β8 l1 1 d1 d3 3 β6 l8 l7 d4 β7 l9 4 l10 l11 L β1 x ROBOTICS: Vladimír Smutný Slide 21, Page 21
22 Typical structure of manipulators Cartezian PPP The body in the space has 6 DOF. The robot shall have at least 6 DOF to allow (even in limited space) any position and orientation of the body. The general manipulators have 6 DOF satisfying this condition, more degrees of freedom means more expensive both in HW and in control algorithms and lower stifness as well. The joints which shall be measured and controlled are mostly prismatic or revolute as the joints with more degrees of freedom are difficult to build. On the other hand the joint which need not be measured and controlled can easily be spherical or cylindrical, as this does not present technical problem. Serial manipulator shall of course have all joints measured and controlled (Why??). Serial manipulator with 6 DOF consisting of prismatic and revolute joints shall have at least three DOF with revolute joints, allowing arbitrary orientation of the manipulated body/accesory. Could you explain why manipulator with 6 prismatic joints cannot orient the body arbitrarily? Typically three first joints (counting from the base), determine the shape and properties of working envelope. This gives rise to the classification of the robot structure according the type of the joints. The list presented is neither exhausting all mathematical combinations nor all structures used in real life. The letters denote the order of the joints, e.g. RPP is revoluteprismatic-prismatic, that is cylindrical robot. ROBOTICS: Vladimír Smutný Slide 22, Page 22
23 Typical structure of manipulators Cylindrical RPP ROBOTICS: Vladimír Smutný Slide 23, Page 23
24 Typical structure of manipulators Spherical RRP ROBOTICS: Vladimír Smutný Slide 24, Page 24
25 Typical structure of manipulators Angular RRR Angular robot has large working space compare to di- mensions. ROBOTICS: Vladimír Smutný Slide 25, Page 25
26 Typical structure of manipulators cranes RRP and RRPP ROBOTICS: Vladimír Smutný Slide 26, Page 26
27 Typical structure of manipulators SCARA RRRP Animations taken from Masud Salimian s web page Scara is particularly suited for operations above horizontal plane, where three vertical axes does not work against gravity. ROBOTICS: Vladimír Smutný Slide 28, Page 27
28 Typical structure of manipulators Stewart/Gough platform ROBOTICS: Vladimír Smutný Slide 29, Page 28
29 Extending the robot reach Angular robot on gantry Systems with more degrees of freedom are used mainly for two reasons: Increasing the volume of work envelope. Robot mounted (usually hanged) on one or two gantry axes could operate much larger space. Getting access to confined areas. Robot with 6 DOF could position rigid body or tool anywhere in the space. 5 DOF are enough for positioning of the rotationally symmetrical welding head. Welds are often hidden in confined places where all solutions of inverse kinematics are colliding with the product. One or two additional rotation axes positioning the product are often used to allow one or more robots mounted on the floor to access whole welding trajectory. ROBOTICS: Vladimír Smutný Slide 30, Page 29
30 Extending the robot reach Angular robots, product on turn table ROBOTICS: Vladimı r Smutny Slide 32, Page 30
31 Extending the robot reach Angular robot, product on turn table ROBOTICS: Vladimı r Smutny Slide 33, Page 31
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