Constraint and velocity analysis of mechanisms

Constraint and velocity analysis of mechanisms Matteo Zoppi Dimiter Zlatanov DIMEC University of Genoa Genoa, Italy Su

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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 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 ZZ-3

Outline Generalities Su ZZ-4

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 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 ZZ-5

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) 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 ZZ-6

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 n 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 ZZ-7

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 the same basis at every configuration is preferable but it is not possible in general It depends on the motion pattern of the mechanism ZZ-8

The spatial case A Plücker basis of twists 3 rotations about, 3 translations along the frame axes A Plücker basis of wrenches 3 pure forces along, 3 moments about the frame axes ZZ-9

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 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 ZZ-10

Outline Generalities Constraint and mobility analysis Su ZZ-11

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 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 ZZ-12

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 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 ZZ-13

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 proceedings I. Bonev, D. Zlatanov, C. Gosselin, Instantaneous kinematics of parallelchain 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 ZZ-14

(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 ZZ-15

Leg and combined freedoms/constraints in PMs Leg freedoms: Leg constraints: End-eff. freedoms: End-eff. constraints: ZZ-16

Outline Generalities Constraint and mobility analysis Examples of geometric constraint and mobility analysis ZZ-17

Example #1.0 Planar mechanisms 3-dof PPMs with identical legs Su ZZ-18

#1.0.1 3-dof 3-RPR PPM Actuation Base R P Su ZZ-19

#1.0.2 3-dof 3-RRR PPM Actuation Base R Mid R Su ZZ-20

#1.0.3 3-dof 3-PRR PPM Actuation P Mid R Su ZZ-21

#1.0.4 3-dof 3-RPP PPM Actuation End-eff. P Mid P Su ZZ-22

#1.0.5 3-dof 3-RRP PPM Actuation Base R Su ZZ-23

#1.0.6 3-dof 3-PRP PPM Actuation Base P Su ZZ-24

Example #1.1 3R1T PM O ZZ-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

#1.1 The 4-5R PM four identical legs first three joint axes in every leg intersecting at a point in the base last two joint axes in every leg parallel to a plane in the platform ZZ-26

#1.1 Constraints and freedoms Screw systems 1. Leg constraint pure force thru O parallel to platform 2. Platform constraints planar pencil of forces parallel to the platform 3. Platform freedoms rotations and 1 translation 4-dof PM ZZ-27

Example #1.2 A 3-CRR mechanism Tripteron Kong and Gosselin, 2002 Screw systems 1. Leg constraints 2 pure moments normal to the joints 2. Platform constraints 3 moments 3. Platform freedoms 3 translations 3-dof PM ZZ-28

Example #1.3 A 3-ERR mechanism Huang and Li, 2002 Screw systems 1. Leg constraints A pure moment normal to all R joints 2. Platform constraints 3 moments 3. Platform freedoms 3 translations 3-dof PM ZZ-29

Example #1.4 A 3-ERR mechanism Huang and Li, 2002 Screw systems 1. Leg constraints A pure force thru O parallel to the 1st R 2. Platform constraints 2 horizontal forces 3. Platform freedoms 3 rotations 1 translation 4-dof PM ZZ-30

Example#1.5 A 3- RR (RRR) mechanism Huang and Li, 2002 Screw systems 1. Leg constraints A pure force vertical thru O 2. Platform constraints 1 vertical force 3. Platform freedoms 3 rotations 2 translations 5-dof PM ZZ-31

Example #1.6 A 3-ERR mechanism Huang and Li, 2002 Screw systems 1. Leg constraints A pure force vertical thru O 2. Platform constraints 1 vertical force 3. Platform freedoms 3 rotations 2 translations 5-dof PM ZZ-32

Example #1.7 A 3-ERR mechanism Huang and Li, 2002 Screw systems 1. Leg constraints A pure moment normal to all R joints 2. Platform constraints 1 vertical moment 3. Platform freedoms 2 rotations 3 translations 5-dof PM ZZ-33

Example #1.8 DYMO 3T Screw systems 1. Leg constraints A pure moment normal to all R joints 2. Platform constraints 3 moments 3. Platform freedoms 3 translations 3-dof translational PM ZZ-34

#1.8 DYMO 3R Screw systems 1. Leg constraints A pure force thru O parallel to middle Rs 2. Platform constraints 3 forces thru O 3. Platform freedoms 3 rotations 3-dof orientational PM ZZ-35

ZZ-36 #1.8 DYMO 3Pl Screw systems 1. Leg constraints A pure force at intersec of extr Rs and to middle Rs [what if extr Rs?] 2. Platform constraints 3 vertical forces 3. Platform freedoms 2 translations 1 rotation 3-dof planar-motion PM

#1.8 DYMO 0 Screw systems 1. Leg constraints A pure force thru O A moment 2. Platform constraints 3 forces thru O 3 moments 3. Platform freedoms zero Platform is locked ZZ-37

#1.8 DYMO 3CVC (constant velocity coupling) Screw systems 1. Leg constraints A pure force in bisecting plane 2. Platform constraints 3 coplanar forces 3. Platform freedoms 2 coplanar rotations 1 normal translation 3-dof CVC PM ZZ-38

Outline Generalities Constraint and mobility analysis Examples of geometric constraint and mobility analysis Velocity equations and Jacobian analysis of PMs ZZ-39

Assumptions We consider a generic PM with any number serial legs labeled The generic L leg comprises 1-dof joints numbered from the base 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 of ZZ-40 [equalities for most configurations/mechanisms]

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 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 ZZ-41

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 is the total structural constraint that the legs apply to the end-eff. ZZ-42

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 a basis of a basis of Note:, the and the space they span are not unique! Note: without singularities and redundancies the are ZZ-43

Input-Output velocity equations The input-output velocity equations are obtained calculating the end-eff. twist along the leg chains These eqs. Contain both active and passive joint velocities The active velocities are assigned The passive velocities are unknown ZZ-44

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 actuated motions ZZ-45

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 In matrix form: Z c is called Jacobian of constraints [preferably chose the as smooth functions of the mechanism configuration] ZZ-46

Jacobian of actuations The equations in the actuated velocities give Square and nonsingular is no singularities or redundancies in the legs The scalar if the leg contains one actuated joint: ZZ-47

Combined equations The actuation and constraint equations can be combined in the form When the matrix at one side is square we can calculate a PM Jacobian ZZ-48

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 the constraint eqs. In this way you simplify rows and columns of the matrices ZZ-49

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 ZZ-50

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 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 ZZ-51

Example #2.0 Planar PMs The velocity eqs can be arranged in the matrix form Finally for every PPM and configuration we have ZZ-52

Example #2.1 3R1T PM Zlatanov-Gosselin, 2001; Zoppi-Zlatanov, 2004 Su ZZ-53

#2.1 Constraints [If no leg is singular] Su ZZ-54

#2.1 Jacobian analysis We take as the 4 The velocity equations are of the type We need symbolic expressions of the [we do not need to work out these 2 components] We eliminate the passive velocities from the velocity eqs by reciprocal product ZZ-55

#2.1 Jacobian analysis We arrange the eqs in matrix form using 6 coordinates A suitable rotating frame is used so 2 rows and columns can be eliminated ZZ-56

Example #2.2 Huang and Li, 2003 PM with five P RR (RR) legs Su ZZ-57

#2.2 Leg constraints The structural constraint of each leg is then the combined structural constraint is The actuated leg constraint is [1-system 5-dof] The combined actuated constraint is ZZ-58

#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 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 ZZ-59

#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 We can suppress the v z coordinate and obtain ZZ-60

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 ZZ-62

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 ZZ-64

#3.1 Leg constraints For the serial legs The structural constraint is spanned by a vertical force thru O The actuated constraints are with an additional force at the intersection of the leg planes ZZ-65

#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) The 2-PRR planar PM imposes the planar constraint and the actuated constraint ZZ-66

#3.1 Leg constraints The combined constraint applied is with the moment in direction The total actuated constraint of the SP leg is ZZ-67

#3.1 Combined constraint The combined structural constraint is The combined actuated constraint is The actuated constraints are a 6-system and the mechanism has 4-dofs commanded by the 4 base P joints ZZ-68

#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 The effect of the interconnection is to change the motion pattern of the mechanism and its dof ZZ-69

#3.1 Velocity equations The velocity eqs can be arranged in matrix form where we use a reference frame to have w x and v z always null ZZ-70

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 ZZ-71

#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 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 ZZ-72

#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 ZZ-73

#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: The combined constraint is direction It is a 3-system Only the wrenches reciprocal to the end-eff. R joint can be transmitted to the end-eff. ZZ-74

#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 [remember that ] The vertical force belongs also to the structural constraint, thus is a 6-system ZZ-75

#3.2 Velocity equations The velocity eqs along one lateral leg and half of the S-P leg are The elimination of the passive velocities is not straightforward in this case ZZ-76

#3.2 Velocity equations We calculate the reciprocal products by respectively We add the resulting eqs and simplify using We obtain the 2 eqs ZZ-77

#3.2 Velocity equations Two more eqs come from the 2 subchains of the S-P leg From which we eliminate the passive velocities in the standard way obtaining ZZ-78

#3.2 Velocity equations The velocity eqs can be arranged in matrix form where we use a reference frame to have w x and v z always null ZZ-79

Example #3.3 ArmillEye IC version of the 3R1T PM used in a previous example Su ZZ-80

#3.3 Leg constraints Legs A and B are serial with 5 joints each With actuated joint free they transmit a pure force With actuated joint locked they transmit wrenches belonging to a 2-system a basis of which contains 2 pure forces ZZ-81

#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 Transmits (as 2 separate serial legs) But due to the interconnection it can transmit additional wrenches [!] ZZ-82

#3.3 Interconnection constraint These interconnection constraints have to be reciprocal to the base joint twist and to belong to the structural constraint So in a nonsingular configuration with ZZ-83

#3.3 Combined constraint The space of the structural constraints is 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 ZZ-84

#3.3 Velocity equations We calculate the end-effector twist along the four leg chains (A,B and C considered as 2 serial) We eliminate the passive joint velocities calculating the reciprocal products with the leg wrenches A,B,CA,CB 4 eqs ZZ-85

#3.3 Matrix form The equations are rearranged in matrix form and expressed interms of the geometry parameters ZZ-86

Example #3.4 Agraule 5-dof ICM with 3 lateral P 2 U 2 S 2 R and 1 central PRUP leg Su ZZ-87

#3.4 Leg constraints Central leg With actuated joints free With actuated joints locked (a planar pencil and a moment) ZZ-88

#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 can transmit a pure force Because the lateral legs are interconnected they can transmit additional constraints ZZ-89

#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 to the R joint At the base side can be transmitted to base if reciprocal to the P joint ZZ-90

#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 And the combined constraint provided by the lateral legs is a 1-system [!] ZZ-91

#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 mechanism has 5-dofs The combined actuated constraint is as with independent legs ZZ-92

#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 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 ZZ-93

#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 These are the structural constraint forces along the US links [!] By means of which we obtain 3 velocity eqs ZZ-94

#3.4 Velocity equations We consider then the central leg 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 ZZ-95

#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 ZZ-96

#3.4 Matrix form The matrices can be expressed using the geometry parameters of the mechanism Su ZZ-97