Constraint and velocity analysis of mechanisms · 2009-10-08 · Summer Screws 2009 Derivation of...
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ummer Screws 2009 mmer Screws 2009 mmer Screws 2009 ummer Screws 2009 Constraint and velocity analysis of mechanisms Su Su Su Su Matteo Zoppi – Dimiter Zlatanov DIMEC – University of Genoa Genoa, Italy ummer Screws 2009 mmer Screws 2009 mmer Screws 2009 ummer Screws 2009 Su Su Su Su Constraint and velocity analysis of mechanisms ZZ-2 ummer Screws 2009 mmer Screws 2009 mmer Screws 2009 ummer Screws 2009 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 Su Su Su Su Constraint and velocity analysis of mechanisms ZZ-3 Examples of Jacobian analysis of PMs 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 ummer Screws 2009 mmer Screws 2009 mmer Screws 2009 ummer Screws 2009 Outline Generalities Su Su Su Su Constraint and velocity analysis of mechanisms ZZ-4
Transcript of Constraint and velocity analysis of mechanisms · 2009-10-08 · Summer Screws 2009 Derivation of...
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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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Constraint and velocity analysis of mechanismsZZ-3
� Examples of Jacobian analysis of PMs
� 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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Constraint and velocity analysis of mechanismsZZ-4
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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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Constraint and velocity analysis of mechanismsZZ-5
� 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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Constraint and velocity analysis of mechanismsZZ-6
� 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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Constraint and velocity analysis of mechanismsZZ-7
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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Constraint and velocity analysis of mechanismsZZ-8
� 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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Constraint and velocity analysis of mechanismsZZ-9
� 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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Constraint and velocity analysis of mechanismsZZ-10
� 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
� Constraint and mobility analysis
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Constraint and velocity analysis of mechanismsZZ-11
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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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Constraint and velocity analysis of mechanismsZZ-12
� 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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Constraint and velocity analysis of mechanismsZZ-13
� 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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Constraint and velocity analysis of mechanismsZZ-14
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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Constraint and velocity analysis of mechanismsZZ-15
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Leg and combined freedoms/constraints in PMs
Leg freedoms:
Leg constraints:
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Constraint and velocity analysis of mechanismsZZ-16
End-eff. freedoms:
End-eff. constraints:
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Outline
� Generalities
� Constraint and mobility analysis
� Examples of geometric constraint and mobility analysis
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Constraint and velocity analysis of mechanismsZZ-17
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Example #1.0 – Planar mechanisms
� 3-dof PPMs with identical legs
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Constraint and velocity analysis of mechanismsZZ-18
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#1.0.1 – 3-dof 3-RPR PPM
� Actuation
� Base R
� P
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Constraint and velocity analysis of mechanismsZZ-19
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#1.0.2 – 3-dof 3-RRR PPM
� Actuation
� Base R
� Mid RS
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Constraint and velocity analysis of mechanismsZZ-20
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#1.0.3 – 3-dof 3-PRR PPM
� Actuation
� P
� Mid R
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Constraint and velocity analysis of mechanismsZZ-21
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#1.0.4 – 3-dof 3-RPP PPM
� Actuation
� End-eff. P
� Mid P
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Constraint and velocity analysis of mechanismsZZ-22
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#1.0.5 – 3-dof 3-RRP PPM
� Actuation
� Base R
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Constraint and velocity analysis of mechanismsZZ-23
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#1.0.6 – 3-dof 3-PRP PPM
� Actuation
� Base P
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Constraint and velocity analysis of mechanismsZZ-24
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Example #1.1 – 3R1T PM
O
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Constraint and velocity analysis of mechanismsZZ-25
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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Constraint and velocity analysis of mechanismsZZ-26
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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Constraint and velocity analysis of mechanismsZZ-27
2. Platform constraints
� 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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Constraint and velocity analysis of mechanismsZZ-28
Tripteron
Kong and Gosselin, 2002
2. Platform constraints
� 3 moments
3. Platform freedoms
� 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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Constraint and velocity analysis of mechanismsZZ-29
Huang and Li, 2002
joints
2. Platform constraints
� 3 moments
3. Platform freedoms
� 3 translations
3-dof PM
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Example #1.4 – A 3-ERR mechanism
Screw systems
1. Leg constraints
� A pure force thru Oparallel to the 1st R
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Constraint and velocity analysis of mechanismsZZ-30
Huang and Li, 2002
2. Platform constraints
� 2 horizontal forces
3. Platform freedoms
� 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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Constraint and velocity analysis of mechanismsZZ-31
Huang and Li, 2002
2. Platform constraints
� 1 vertical force
3. Platform freedoms
� 3 rotations
� 2 translations
5-dof PM
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Example #1.6 – A 3-ERR mechanism
Screw systems
1. Leg constraints
� A pure force vertical thru O
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Constraint and velocity analysis of mechanismsZZ-32
Huang and Li, 2002
2. Platform constraints
� 1 vertical force
3. Platform freedoms
� 3 rotations
� 2 translations
5-dof PM
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Example #1.7 – A 3-ERR mechanism
Screw systems
1. Leg constraints
� A pure moment normal to all R joints
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Constraint and velocity analysis of mechanismsZZ-33
Huang and Li, 2002
2. Platform constraints
� 1 vertical moment
3. Platform freedoms
� 2 rotations
� 3 translations
5-dof PM
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Example #1.8 – DYMO 3T
Screw systems
1. Leg constraints
� A pure moment normal to all R joints
Platform constraintsSum
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Constraint and velocity analysis of mechanismsZZ-34
2. Platform constraints
� 3 moments
3. Platform freedoms
� 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
Platform constraintsSum
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Constraint and velocity analysis of mechanismsZZ-35
2. Platform constraints
� 3 forces thru O
3. Platform freedoms
� 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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Constraint and velocity analysis of mechanismsZZ-36
[what if extr Rs ||?]
2. Platform constraints
� 3 vertical forces
3. Platform freedoms
� 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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Constraint and velocity analysis of mechanismsZZ-37
2. Platform constraints
� 3 forces thru O
� 3 moments
3. Platform freedoms
� 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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Constraint and velocity analysis of mechanismsZZ-38
2. Platform constraints
� 3 coplanar forces
3. Platform freedoms
� 2 coplanar rotations
� 1 normal translation
3-dof CVC PM
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Outline
� Generalities
� Constraint and mobility analysis
� Examples of geometric constraint and mobility analysis
� Velocity equations and Jacobian analysis of PMs
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Constraint and velocity analysis of mechanismsZZ-39
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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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Constraint and velocity analysis of mechanismsZZ-40
� 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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Constraint and velocity analysis of mechanismsZZ-41
� 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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Constraint and velocity analysis of mechanismsZZ-42
� 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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Constraint and velocity analysis of mechanismsZZ-43
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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Constraint and velocity analysis of mechanismsZZ-44
� 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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Constraint and velocity analysis of mechanismsZZ-45
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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Constraint and velocity analysis of mechanismsZZ-46
� 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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Constraint and velocity analysis of mechanismsZZ-47
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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Constraint and velocity analysis of mechanismsZZ-48
� 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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Constraint and velocity analysis of mechanismsZZ-49
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
� 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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Constraint and velocity analysis of mechanismsZZ-50
� Examples of Jacobian analysis of PMs
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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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Constraint and velocity analysis of mechanismsZZ-51
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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Constraint and velocity analysis of mechanismsZZ-52
� 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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Constraint and velocity analysis of mechanismsZZ-53
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#2.1 – Constraints[If no leg is singular]
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Constraint and velocity analysis of mechanismsZZ-54
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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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Constraint and velocity analysis of mechanismsZZ-55
� 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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Constraint and velocity analysis of mechanismsZZ-56
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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Constraint and velocity analysis of mechanismsZZ-57
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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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Constraint and velocity analysis of mechanismsZZ-58
[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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Constraint and velocity analysis of mechanismsZZ-59
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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#2.2 – Velocity equations
� 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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Constraint and velocity analysis of mechanismsZZ-60
� Due to the structural constraint
� We can suppress the vz coordinate and obtain
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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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Constraint and velocity analysis of mechanismsZZ-61
� Examples of Jacobian analysis of PMs
� Extension to non purely parallel mechanisms: S-PMs and ICMs
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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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Constraint and velocity analysis of mechanismsZZ-62
belonging to different in-parallel chains
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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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Constraint and velocity analysis of mechanismsZZ-63
� Examples of Jacobian analysis of PMs
� 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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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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Constraint and velocity analysis of mechanismsZZ-64
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#3.1 – Leg constraints
� For the serial legs
� The structural constraint is spanned by a vertical force thru O
� The actuated constraints are
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Constraint and velocity analysis of mechanismsZZ-65
� The actuated constraints are
with an additional force at the intersection of the leg planes
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#3.1 – Leg constraints
� 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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Constraint and velocity analysis of mechanismsZZ-66
� 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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Constraint and velocity analysis of mechanismsZZ-67
� The total actuated constraint of the SP leg is Sum
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#3.1 – Combined constraint
� The combined structural constraint is
� The combined actuated constraint is
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Constraint and velocity analysis of mechanismsZZ-68
� The actuated constraints are a 6-system and the mechanism has 4-dofs commanded by the 4 base P joints
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#3.1 – Velocity equations
� 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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Constraint and velocity analysis of mechanismsZZ-69
interconnection
� The effect of the interconnection is to change the motion pattern of the mechanism and its dof
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#3.1 – Velocity equations
� The velocity eqs can be arranged in matrix form
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Constraint and velocity analysis of mechanismsZZ-70
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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Constraint and velocity analysis of mechanismsZZ-71
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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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Constraint and velocity analysis of mechanismsZZ-72
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
� The actuated constraints are
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Constraint and velocity analysis of mechanismsZZ-73
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#3.2 – Actuated constraints
� 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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Constraint and velocity analysis of mechanismsZZ-74
� 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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#3.2 – Actuated constraints
� 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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Constraint and velocity analysis of mechanismsZZ-75
[remember that ]
� The vertical force belongs also to the structural constraint, thus
�
� is a 6-system
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#3.2 – Velocity equations
� The velocity eqs along one lateral leg and half of the S-P leg are
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Constraint and velocity analysis of mechanismsZZ-76
� The elimination of the passive velocities is not straightforward in this case
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#3.2 – Velocity equations
� We calculate the reciprocal products by respectively
� We add the resulting eqs and simplify using
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Constraint and velocity analysis of mechanismsZZ-77
� We obtain the 2 eqs
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#3.2 – Velocity equations
� Two more eqs come from the 2 subchains of the S-P leg
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Constraint and velocity analysis of mechanismsZZ-78
� From which we eliminate the passive velocities in the standard way obtaining
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#3.2 – Velocity equations
� The velocity eqs can be arranged in matrix form
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Constraint and velocity analysis of mechanismsZZ-79
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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Constraint and velocity analysis of mechanismsZZ-80
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#3.3 – Leg constraints
� Legs A and B are serial with 5 joints each
� With actuated joint free they transmit a pure force
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Constraint and velocity analysis of mechanismsZZ-81
� With actuated joint locked they transmit wrenches belonging to a 2-system a basis of which contains 2 pure forces
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#3.3 – Leg constraints
� 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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Constraint and velocity analysis of mechanismsZZ-82
� 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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Constraint and velocity analysis of mechanismsZZ-83
� So in a nonsingular configuration
with
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#3.3 – Combined constraint
� The space of the structural constraints is
� The space of the actuated constraints is
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Constraint and velocity analysis of mechanismsZZ-84
� 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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#3.3 – Velocity equations
� We calculate the end-effector twist along the four leg chains (A,B and C considered as 2 serial)
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Constraint and velocity analysis of mechanismsZZ-85
� 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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Constraint and velocity analysis of mechanismsZZ-86
�
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Example #3.4 – Agraule
� 5-dof ICM with 3 ‘lateral’ P2U2S2R and 1 ‘central’ PRUP leg
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Constraint and velocity analysis of mechanismsZZ-87
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#3.4 – Leg constraints
Central leg
� With actuated joints freeS
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Constraint and velocity analysis of mechanismsZZ-88
� With actuated joints locked
(a planar pencil and a moment)
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#3.4 – Leg constraints
� 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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Constraint and velocity analysis of mechanismsZZ-89
� 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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#3.4 – Interconnection constraint
� 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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Constraint and velocity analysis of mechanismsZZ-90
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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#3.4 – Interconnection constraint
� 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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Constraint and velocity analysis of mechanismsZZ-91
� Out of singularities a solution exists
� And the combined constraint provided by the lateral legs is a 1-system [!]
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#3.4 – Combined constraint
� 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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Constraint and velocity analysis of mechanismsZZ-92
� 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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#3.4 – Velocity equations
� 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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Constraint and velocity analysis of mechanismsZZ-93
� 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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#3.4 – Velocity equations
� 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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Constraint and velocity analysis of mechanismsZZ-94
� These are the structural constraint forces along the US links [!]
� By means of which we obtain 3 velocity eqs
�
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#3.4 – Velocity equations
� We consider then the central leg
� 2 actuated joints � we have a moment reciprocal
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Constraint and velocity analysis of mechanismsZZ-95
� 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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#3.4 – Matrix form
� 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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Constraint and velocity analysis of mechanismsZZ-96
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#3.4 – Matrix form
� The matrices can be expressed using the geometry parameters of the mechanism
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Constraint and velocity analysis of mechanismsZZ-97
