Summer Screws 2009 - unige.it › ... › SS09 › PDF › Part3 › SS09-Zoppi.pdf · 2009-10-08 · Summer Screws 2009 Mechanisms with Dimension of the problem n

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Constraint and velocity analysis of mechanisms

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Matteo Zoppi – Dimiter Zlatanov

DIMEC – University of Genoa

Genoa, Italy

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Constraint and velocity analysis of mechanismsZZ-2

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Outline

� Generalities

� Constraint and mobility analysis

� Examples of geometric constraint and mobility analysis

� Velocity equations and Jacobian analysis of PMs

� Examples of Jacobian analysis of PMs

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� Extension to non purely parallel mechanisms: S-PMs and ICMs

� Example of constraint and velocity analysis of an S-PM

� Examples of constraint and velocity analysis of ICMs

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Outline

� Generalities

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Derivation of I/O velocity equations

The conventional process of deriving the input-output velocity equation for a parallel mechanism consists in differentiating the inverse kinematic equations

� Generally a tedious process

� Possible parameterisation errors (motion pattern and

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� Possible parameterisation errors (motion pattern and singularities)

A much better approach is the use of reciprocal screws

� Better geometrical insight into the problem

� Easier precise and complete description of singularity types

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Dimension of the problem

Mechanisms with 6 DOFs� it is expected that twists, wrenches and the velocity equations have dimension 6

Mechanisms with n<6 DOFs� It is desirable to treat twists (instantaneous motions)

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� It is desirable to treat twists (instantaneous motions) and wrenches (forces and moments) in the velocity and singularity analysis as n-dimensional

� The matrices involved are desirably nxn

� The coordinate system in which this is possible depends on the motion pattern and may vary with the configuration

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Dimension of the problem

Mechanisms with n<6 DOFs� The velocity analysis amounts to an n-dimensional version of screw calculus

� Screws and reciprocal screws (i.e., twists and wrenches) in general have different sets of ncoordinates

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coordinates

� Unlike the general 6-DOF case, screws and reciprocal screws can no longer be thought of as elements of the same vector space

A particular class are planar mechanisms� Three-dimensional planar screws

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Screw basis

� The twist/wrench basis used for the description

� Must have a maximum number of independent reciprocal screws at every configuration

� this number may change at singular configurations

� May change with the configuration

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� May change with the configuration

� the same basis at every configuration is preferable but it is not possible in general

� It depends on the motion pattern of the mechanism

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The spatial case

� A Plücker basis of twists

3 rotations about, 3 translations along the frame axes

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� 3 rotations about, 3 translations along the frame axes

� A Plücker basis of wrenches

� 3 pure forces along, 3 moments about the frame axes

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The planar case

� Origin and x-y axes in the plane of motion

� The twist system of planar motion is

� Planar twists have always equal to zero

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� The wrench of planar actuations can be

[no interest in considering wrenches that are reciprocal to every planar twist]

� The reciprocal product of planar twists/wrenches

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Outline

� Generalities


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The screw-theory method for velocity analysis of PMs

Overview

� Write a system of velocity equations along the leg chains – These equations contain both active and passive joint velocities

� The active joint velocities are assigned

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� The passive joint velocities are unknown

� The output velocities (end-effector twist) are the goal

� Eliminate the passive joint velocities using a screw-theory method

� Obtain a system of linear input-output velocity equations containing only the active joint velocities

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Historical #1� The origins of the method can be found in

� K. H. Hunt, Kinematic Geometry of Mechanisms, Oxford University Press, 1978

� It was first presented in� M. Mohamed, J. Duffy, A direct determination of the instantaneous

kinematics of fully parallel robot manipulators, in: ASME Design Eng. Techn. Conf., 1984, pp. ASME paper 83–DET–114

� M. Mohamed, J. Duffy, A direct determination of the instantaneous

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� M. Mohamed, J. Duffy, A direct determination of the instantaneous kinematics of fully parallel robot manipulators, ASME J. of Mechanisms, Transmissions and Automation in Design 107 (2) (1985) 226–229

� It was then developed in� V. Kumar, Instantaneous kinematics of parallel-chain robotic

mechanisms, in: ASME 21th Mechanisms Conference, Mechanism Synthesis and Analysis, 1990, pp. 279–287

� V. Kumar, Instantaneous kinematics of parallel-chain robotic mechanisms, ASME JMD 114(3) (1992) 349–358

� S. Agrawal, Rate kinematics of in-parallel manipulator systems, in: IEEE ICRA90, 1990, pp. 104–109 vol.1

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Historical #2� Cases with *more than one actuated joint per leg and *limited-dof

with identical leg constraints in� D. Zlatanov, B. Benhabib, R. Fenton, Velocity and singularity analysis of

hybrid chain manipulators, in: ASME 23rd Biennial Mechaism Conference in DETC94, Vol. 70, Minneapolis, MN, USA, 1994, pp. 467–476

� The application to planar PMs is discussed in particular in� K. Hunt, Don’t cross-thread the screw, in: Ball-2000 Symposium,

University of Cambridge at Trinity College, Cambridge, 2000, CD

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University of Cambridge at Trinity College, Cambridge, 2000, CD proceedings

� I. Bonev, D. Zlatanov, C. Gosselin, Instantaneous kinematics of parallel-chain robotic mechanisms, ASME JMD 125 (3) (2003) 573–581

� A generalization to any number of actuated joints in the legs and the discussion of non purely parallel mechanisms in� M. Zoppi, D. Zlatanov, and R. Molfino. On the velocity analysis of

interconnected chains mechanisms. Int. J. Mech. and Machine Theory, 41(11):1346-1358, 2006.

� See also Joshi, Tsai. Jacobian Analysis of Limited-DOF Parallel Manipulators. ASME JMD 124(2), 2002

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(Purely) Parallel Mechanisms (PMs) –recall–

� Composed of an end-effector connected to the base by independent, serial leg chains

� Any leg architecture

� Any number of actuated joints in each leg

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Leg and combined freedoms/constraints in PMs

Leg freedoms:

Leg constraints:

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End-eff. freedoms:

End-eff. constraints:

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Outline

� Generalities



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Example #1.0 – Planar mechanisms

� 3-dof PPMs with identical legs

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#1.0.1 – 3-dof 3-RPR PPM

� Actuation

� Base R

� P

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#1.0.2 – 3-dof 3-RRR PPM

� Actuation

� Base R

� Mid R

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#1.0.3 – 3-dof 3-PRR PPM

� Actuation

� P

� Mid R

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#1.0.4 – 3-dof 3-RPP PPM

� Actuation

� End-eff. P

� Mid P

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#1.0.5 – 3-dof 3-RRP PPM

� Actuation

� Base R

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#1.0.6 – 3-dof 3-PRP PPM

� Actuation

� Base P

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Example #1.1 – 3R1T PM

O

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Zlatanov and Gosselin, 2001

� the first three joint axes

� intersect at O

� the last two joint axes

� are parallel

� point O

� fixed in the base

� common to all legs

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#1.1 – The 4-5R PM

� four identical legs

� first three joint axes in every leg

� intersecting at a point in the base

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point in the base

� last two joint axes in every leg

� parallel to a plane in the platform

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#1.1 – Constraints and freedoms

Screw systems

1. Leg constraint

� pure force thru O parallel to platform

2. Platform constraints

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� planar pencil of forces parallel to the platform

3. Platform freedoms

� rotations and 1 translation

4-dof PM

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Screw systems

1. Leg constraints

� 2 pure moments normal to the joints

Platform constraints

Example #1.2 – A 3-CRR mechanism

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Tripteron

Kong and Gosselin, 2002


� 3 moments


� 3 translations

3-dof PM

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Example #1.3 – A 3-ERR mechanism

Screw systems

1. Leg constraints

� A pure moment normal to all R joints

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Huang and Li, 2002

joints


� 3 moments


� 3 translations

3-dof PM

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Screw systems

1. Leg constraints

� A pure force thru Oparallel to the 1st R

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Huang and Li, 2002


� 2 horizontal forces


� 3 rotations

� 1 translation

4-dof PM

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Example#1.5– A 3-|RR|(RRR) mechanism

Screw systems

1. Leg constraints

� A pure force vertical thru O


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Huang and Li, 2002


� 1 vertical force


� 3 rotations

� 2 translations

5-dof PM

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Screw systems

1. Leg constraints

� A pure force vertical thru O


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Huang and Li, 2002


� 1 vertical force


� 3 rotations

� 2 translations

5-dof PM

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Screw systems

1. Leg constraints



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Huang and Li, 2002


� 1 vertical moment


� 2 rotations

� 3 translations

5-dof PM

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Example #1.8 – DYMO 3T

Screw systems

1. Leg constraints



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� 3 moments


� 3 translations

3-dof translational PM

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#1.8 – DYMO 3R

Screw systems

1. Leg constraints

� A pure force thru Oparallel to middle Rs


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� 3 forces thru O


� 3 rotations

3-dof orientational PM

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#1.8 – DYMO 3PlScrew systems

1. Leg constraints

� A pure force at intersec of extr Rs and || to middle Rs[what if extr Rs ||?]

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[what if extr Rs ||?]


� 3 vertical forces


� 2 translations

� 1 rotation

3-dof planar-motion PM

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#1.8 – DYMO 0

Screw systems

1. Leg constraints

� A pure force thru O

� A moment

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� 3 forces thru O

� 3 moments


� zero

Platform is locked

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#1.8 – DYMO 3CVC (constant velocity coupling)

Screw systems

1. Leg constraints

� A pure force in bisecting plane


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� 3 coplanar forces


� 2 coplanar rotations

� 1 normal translation

3-dof CVC PM

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Outline

� Generalities




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Assumptions

� We consider a generic PM with any number of

serial legs labeled

� The generic L leg comprises 1-dof joints numbered from the base

� is the number of actuated joints (>=0)

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� is the number of actuated joints (>=0)

� is the number of passive joints

� is the system spanned by the active joint twists

� is the system spanned by the passive joint twists

� is the system spanned by all joint twists

� We assume legs containing actuated joints

� [equalities for ‘most’ configurations/mechanisms]

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Assumptions

� Two systems of wrenches introduced for each leg:

� is the system of structural constraints� Consists of wrenches reciprocal to all the joint screws

Spans the generalized forces that the leg can transmit

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� Spans the generalized forces that the leg can transmit from end-eff. to base when all joints are free to move

� is the system of actuated constraints� Consists of wrenches reciprocal to the passive joint screws

� Spans the generalized forces that the leg can transmit from end-eff. to base with the actuated joints locked

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End-eff. constraints and mobility

� Every feasible motion of the end-eff. belongs to

� Since all legs are connected to the same end-eff.

� All feasible end-eff. twists must belong to

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� is the total structural constraint that the legs apply to the end-eff.

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End-eff. constraints and mobility

� Due to the different dimensions of the systems of structural and actuated constraints

� we can complete a basis of the structural constraints

with additional wrenches to obtain a basis of the actuated constraints

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actuated constraints

a basis of

a basis of

Note: , the and the space they span are not unique!

Note: without singularities and redundancies the are

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Input-Output velocity equations

� The input-output velocity equations are obtained calculating the end-eff. twist along the leg chains

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� These eqs. Contain both active and passive joint velocities

� The active velocities are assigned

� The passive velocities are unknown

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I-O eqs: elimination of passive velocities

� For each leg

� We take the reciprocal product of each velocity eq. with the wrenches in a basis of

constrained motions

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actuated motions

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Jacobian of constraints

� We do the same for all legs and obtain

� eqs in input velocities

� Any I-O feasible motion satisfies these eqs

� The end-eff. freedom is defined by

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� In matrix form:

� Zc is called Jacobian of constraints[preferably chose the as smooth functions of the mechanism configuration]

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Jacobian of actuations

� The equations in the actuated velocities give

�

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Square and nonsingular is no singularities or redundancies in the legs

The scalar if the leg contains one actuated joint:

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Combined equations

� The actuation and constraint equations can be combined in the form

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� When the matrix at one side is square we can calculate a PM Jacobian

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Combined equations

� Attention to the selection of the reference frame

� The eqs may simplify

� The dimension of the problem may reduce

� Consider reference frames where some coordinates of the end-eff. twist are null due to

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coordinates of the end-eff. twist are null due to the constraint eqs.

� In this way you simplify rows and columns of the matrices

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Outline

� Generalities





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Example #2.0 – Planar PMs

� The end-eff. twist is calculated along each leg

� Each leg with actuated joint locked transmits a planar wrench reciprocal to all joints but the one actuated � we use it to eliminate the passive

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actuated � we use it to eliminate the passive

joint velocities from the velocity eqs

� If a leg has an actuated wrench system of dimension 2 or 3

� More elements in any basis � more eqs

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Example #2.0 – Planar PMs

� The velocity eqs can be arranged in the matrix form

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� Finally for every PPM and configuration we have

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Example #2.1 – 3R1T PM

� Zlatanov-Gosselin, 2001; Zoppi-Zlatanov, 2004

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#2.1 – Constraints[If no leg is singular]

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#2.1 – Jacobian analysis

� We take as the 4

� The velocity equations are of the type

� We need symbolic expressions of the

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� We eliminate the passive velocities from the velocity eqs by reciprocal product

[we do not need to work out these 2 components]

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#2.1 – Jacobian analysis

� We arrange the eqs in matrix form using 6 coordinates

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A suitable rotating frame is used so 2 rows and columns can be eliminated

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Example #2.2 – Huang and Li, 2003

� PM with five P|RR|(RR) legs

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#2.2 – Leg constraints

� The structural constraint of each leg is

� then the combined structural constraint is

[1-system 5-dof]

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[1-system � 5-dof]

� The actuated leg constraint is

� The combined actuated constraint is

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#2.2 – Velocity equations

� We write the end-eff. twist along the different leg chains and obtain the velocity eqs

� We have a non-unique actuation system for each leg

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leg

� The reciprocal product of any basis of the actuation system eliminates the passive velocities

� The eqs (in the active velocities only) are arranged in the matrix form

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� The screws used are expressed using the geometry parameters of the mechanism in order to obtain expressions that can be calculated

Due to the structural constraint

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� Due to the structural constraint

� We can suppress the vz coordinate and obtain

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Outline

� Generalities





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Extension to non purely parallel mechanisms

� A method to obtain the I/O velocity equations in the active joint velocities for mechanisms with any architecture does not exist

� The method can be extended to other classes of architectures derived from purely parallel, in particular� Series-parallel – where individual joints are replaced by parallel

subchains� Interconnected chains – where subchains are added between links

belonging to different in-parallel chains

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belonging to different in-parallel chains

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Outline

� Generalities





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� Example of constraint and velocity analysis of an S-PM

� Examples of constraint and velocity analysis of ICMs

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Example #3.1 – 4-dof 2R2T S-PM� S-PM obtained from (Huang and Li, 2003) by welding one

to the other the 3rd links of two legs

� The new mostly-serial leg comprises a planar PM and a spherical 4-bar linkage

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� For the serial legs

� The structural constraint is spanned by a vertical force thru O

� The actuated constraints are

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with an additional force at the intersection of the leg planes

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� For the mostly-serial leg

� The spherical 4-bar is passive and 1-dof

Its structural constraints are spanned by any 3 forces thru O and 2 moments (each one normal to 2 of the R joints)

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� The 2-PRR planar PM imposes the planar constraint and the actuated constraint

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#3.1 – Leg constraints� The combined constraint applied is

with the moment in direction

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� The total actuated constraint of the SP leg is

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#3.1 – Combined constraint

� The combined structural constraint is

� The combined actuated constraint is

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� The actuated constraints are a 6-system and the mechanism has 4-dofs commanded by the 4 base P joints

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� Locking any actuated joint adds to the end-eff. constraint a force as in the original PM

� We can then write 4 equations expressing the end-eff. twist along the 4 legs disregarding the interconnection

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interconnection

� The effect of the interconnection is to change the motion pattern of the mechanism and its dof

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� The velocity eqs can be arranged in matrix form

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where we use a reference frame

to have wxand v

zalways null

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Example #3.2 – 4-dof 2R2T ICM� ICM obtained modifying the S-PM: the end joints of the 2

serial legs are moved from the platform to 2 opposite links of the spherical 4-bar

� The actuated joints are still the base Ps

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#3.2 – Structural constraints

� The S-P leg (without considering the effect of the others) applies to the end-eff.

� Each serial leg applies to the link of the S-P leg the same vertical force thru O which is also

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the same vertical force thru O which is also reciprocal to the end-eff. R joint of the S-P leg

� Thus the structural constraint is

� The mechanism has the same 4-dof as the S-PM

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#3.2 – Actuated constraints

� Consider first the base joints of the S-P leg locked and the base joints of the lateral legs free

� It is like the lateral legs are not there


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� Lock now the base joint of one of the lateral legs and consider the constraint on the corresponding link of the S-P leg

� From the lateral leg:

From the S-P leg:

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� From the S-P leg:

� The combined constraint is

� It is a 3-system – Only the wrenches reciprocal to the end-eff. R joint can be transmitted to the end-eff.

direction

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� is a cylindroid

� We need a basis of it

� We can take the vertical force thru O and a wrench obtained by the linear combination

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[remember that ]

� The vertical force belongs also to the structural constraint, thus

�

� is a 6-system

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� The velocity eqs along one lateral leg and half of the S-P leg are

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� The elimination of the passive velocities is not straightforward in this case

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� We calculate the reciprocal products by respectively

� We add the resulting eqs and simplify using

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� We obtain the 2 eqs

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� Two more eqs come from the 2 subchains of the S-P leg

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� From which we eliminate the passive velocities in the standard way obtaining

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� The velocity eqs can be arranged in matrix form

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where we use a reference frame

to have wxand v

zalways null

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Example #3.3 – ArmillEye

� IC version of the 3R1T PM used in a previous example

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� Legs A and B are serial with 5 joints each

� With actuated joint free they transmit a pure force

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� With actuated joint locked they transmit wrenches belonging to a 2-system a basis of which contains 2 pure forces

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� Leg C with actuated joint locked

� is equivalent to 2 independent serial legs of type A,B

� Leg C with actuated joint free

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� Leg C with actuated joint free

� Transmits (as 2 separate serial legs)

� But due to theinterconnection it can transmitadditional wrenches [!]

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#3.3 – Interconnection constraint

� These interconnection constraints have to be reciprocal to the base joint twist and to belong to the structural constraint

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� So in a nonsingular configuration

with

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� The space of the structural constraints is

� The space of the actuated constraints is

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� The space of the actuated constraints is

� Out of singularities

Note that we use 4 coordinates because we want to use the same reference frame at every configuration – otherwise 3 are enough

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� We calculate the end-effector twist along the four leg chains (A,B and C considered as 2 serial)

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� We eliminate the passive joint velocities calculating the reciprocal products with the leg wrenches A,B,CA,CB � 4 eqs

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#3.3 – Matrix form

� The equations are rearranged in matrix form and expressed interms of the geometry parameters

�

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�

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Example #3.4 – Agraule

� 5-dof ICM with 3 ‘lateral’ P2U2S2R and 1 ‘central’ PRUP leg

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Central leg

� With actuated joints free

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� With actuated joints locked

(a planar pencil and a moment)

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� First the leg is considered separately from the rest of the mechanism� With base joints free no constraint on the end-effector

With base joints locked the leg

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� With base joints locked the leg can transmit a pure force

� Because the lateral legs are interconnected they can transmit additional constraints

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� Forces transmitted along the US links

� A resultant of these forces

� At the end-effector side can be transmitted to base if reciprocal

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transmitted to base if reciprocal to the R joint

� At the base side can be transmitted to base if reciprocal to the P joint

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� So with free actuators the 6 forces along the US links have to satisfy a system of 6 linear homogeneous equations to be transmitted to base

Out of singularities a solution exists

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� Out of singularities a solution exists

� And the combined constraint provided by the lateral legs is a 1-system [!]

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� The combined structural constraint comprises the interconnection constraint and the constraint of the central leg

Out of singularities the dimension is 1 and the

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� Out of singularities the dimension is 1 and the mechanism has 5-dofs

� The combined actuated constraint is as with independent legs

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� The end-effector twist is calculated along each leg

� We start from the end-eff. along the lateral legs

Twist of the link adjacent to the end-effector

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� ξ ξ ξ ξ L is calculated along the PUS chains (2 eqs leg)

� This time eliminating the passive joint velocities is not immediate as with independent serial legs

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� The leg chain is not serial and no wrench is reciprocal to all the passive joints

� We need 2 wrenches reciprocal to

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� These are the structural constraint forces along the US links [!]

� By means of which we obtain 3 velocity eqs

�

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� We consider then the central leg

� 2 actuated joints � we have a moment reciprocal

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� 2 actuated joints � we have a moment reciprocal

to all joints except the actuated R and a force reciprocal to all but the actuated P

� We multiply alternatively obtaining 2 velocity equations

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� We rearrange the 5 velocity eqs in matrix form

� We use a reference frame with to have the x component of the trans velocity zero

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� The matrices can be expressed using the geometry parameters of the mechanism

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Documents

Summer Screws 2009 - unige.it › ... › SS09 › PDF › Part3 › SS09-Zoppi.pdf · 2009-10-08 · Summer Screws 2009 Mechanisms with Dimension of the problem n