Screw Theory and its Applications in Robotics

Transcription

1 Screw Theory and its Applications in Robotics Marco Carricato Group of Robotics, Automation and Biomechanics University of Bologna Italy IFAC 2017 World Congress, Toulouse, France

2 Table of Contents 1. First-order Rigid-Body Kinematics 2. Statics of Rigid Bodies 3. Freedoms and Constraints 4. Screw Systems 5. Invariant and Persistent Screw Systems 6. Applications Mobility Analysis 7. Application Design 8. Application Singularity Analysis 9. Conclusions

3 First-order Rigid-Body Kinematics 3/72 Twist The body instantaneous motion, or twist, is the combination of a translation along and a rotation about the instantaneous screw axis (ISA), with translational and angular velocities being v = v e and ω = ωe, respectively. The most general twist is an instantaneous helicoidal motion with pitch h = v /ω and amplitude ω. v O = v }{{} e + ω r }{{} e = ω(he + e r)

4 First-order Rigid-Body Kinematics 4/72 v = 0: all points of the ISA are instantaneously stationary, i.e. the body instantaneously rotates about l. ω = 0: all points of the body share the same velocity v, i.e. the body instantaneously translates parallel to e.

5 First-order Rigid-Body Kinematics 5/72 The pair (ω, v O ) completely defines the body velocity field. The Plücker representation of the body twist with respect to an arbitrarily-chosen reference point O is: [ ] [ ] [ ] ω ω e ξ = = = ω = ωσ v O v + r ω he + r e }{{} σ h = v ω = v O ω ω 2 σ is called a screw of pitch h along line l. The twist is completely defined by the triplet (ω, h, l), namely by the screw σ and the amplitude ω.

6 First-order Rigid-Body Kinematics 6/72 Finite-pitch twist associated with a helicoidal joint: [ ω ξ = vo] [ = ω e he + r e ] ω 0, h 0 0-pitch twist associated with a revolute joint: [ ] ω ξ = v O [ ] e = ω r e ω 0, h = 0

7 First-order Rigid-Body Kinematics 7/72 -pitch twist associated with a prismatic joint: [ ω ξ = vo] [ 0 = v e] ω = 0, h =

8 First-order Rigid-Body Kinematics 8/72 Twist addition: 2 Angular velocity of body 3 w.r.t. 1: 1 3 ω 31 = ω 21 + ω 32 Instantaneous motion of body 2 w.r.t. body 1: [ ] [ ] ω21 ω ξ 21 = = 21 v O,21 v 21 + r 1 ω 21 Instantaneous motion of body 3 w.r.t. body 2: [ ] [ ] ω32 ω ξ 32 = = 32 v O,32 v 32 + r 2 ω 32

9 First-order Rigid-Body Kinematics 9/72 Instantaneous motion [ of] body[ 3 w.r.t. body] 1: ω31 ω21 + ω ξ 31 = = 32 = ξ v O,31 v O,21 + v 21 + ξ 32 O,32 Twists form a vector space, called se(3): scalar multiplication: proportional [ increase] of the amplitude e λξ = λω he + r e vector addition: resultant instantaneous motion of a serial connection ξ 21 + ξ 23 = ξ 31 Remarks A vector space must be closed under multiplication and vector addition! The concept of linear dependence or independence only applies to the elements of a vector space.

10 First-order Rigid-Body Kinematics 10/72 Example: Instantaneous spatial rotations The set of spatial pure rotations does not form, in general, a vector space: ξ 1 = [ ] [ ] ω1 ω, ξ 0 2 = 2 r 2 ω 2 [ ] [ ] ω1 + ω ξ 3 = ξ 1 + ξ 2 = 2 ω3 = r 2 ω 2 v 3 h 3 = v 3 ω 3 ω 2 3 = ω 1 (r 2 ω 2 ) 0 The set of spatial rotations through a point forms a vector space (instantaneous spherical motion): ξ 1 = [ ] ω1, ξ 0 2 = [ ] ω2 0 ξ 3 = ξ 1 + ξ 2 = [ ] ω1 + ω 2 0

11 First-order Rigid-Body Kinematics 11/72 The twist of the end-effector of a serial chain µ µ µ [ ] [ ] e ξ EE = ξ i = ω i σ i = ω i e i = i he i +r i e i he i +r i e i i=1 i=1 i=1 }{{} J is the Jacobian matrix of the chain. J ω 1. ω µ

12 First-order Rigid-Body Kinematics 12/72 Twist space of a serial chain ξ EE results from the linear combination of the joint screws σ 1,..., σ µ, with coefficients ω 1,..., ω µ. By varying ω 1,..., ω µ in all possible ways, the joint screws ξ 1,..., ξ µ span a vector subspace T embedded in se(3): T = span {ξ 1,..., ξ µ } = span {σ 1,..., σ µ } T contains all feasible instantaneous motions of the EE. Clearly: dim T = rank(j) If ξ 1,..., ξ µ are linearly independent, dim T = µ; otherwise: dim T < µ.

13 Statics of Rigid Bodies 13/72 Wrench Any system of forces and couples applied to a rigid body is equivalent to a wrench, i.e. a force f applied to the central axis (CA) l and a couple τ parallel to f: The pitch of the wrench is: h = τ f = τ f f 2

14 Statics of Rigid Bodies 14/72 The resultant moment about an arbitrarily-chosen reference point O is: τ O = τ }{{} e + r f = f(he + r e) }{{} e The wrench is completely defined by the pair (f, τ O ) or the triplet (f, h, l). Special cases: the wrench is a pure force: τ = 0, f 0, h = 0 the wrench is a pure couple: τ 0, f = 0, h =

15 Statics of Rigid Bodies 15/72 Plücker representation of a wrench w.r.t. point O: [ ] ( ) [ f axis ζ = = τo coord. τ [ ] ( ) τo ray ˆζ = f coord. ] [ ] f e = f = fσ + r f he + r e

16 Statics of Rigid Bodies 16/72 Wrench addition f = f 1 + f 2 τ O = r 1 f 1 + τ 1 + r 2 f 2 + τ 2 [ ] [ ] f ζ 1 + ζ 2 = 1 f + 2 r 1 f 1 + τ r 2 f 2 + τ 2 [ ] f = Summing the wrenches acting in parallel on a rigid body yields the overall wrench applied to the body. Wrenches form a vector space dual to se(3), called se (3). τ O

17 Statics of Rigid Bodies 17/72 Power: Plücker coordinates of ξ and ζ w.r.t. O: [ [ ] ω ξ =, vo] ˆξ vo = ω [ f ζ = τ O ], ˆζ = [ ] τo f The power developed by the wrench ζ acting on a body moving with twist ξ is given by the reciprocal product between ζ and ξ: P = f v O + τ O ω = ζ ξ = ˆζ ξ = ˆζ T ξ = [ τ T O f ] [ ] T ω = τ T O ω + f T v v O O A twist ξ and a wrench ζ are reciprocal if ζ develops no power on ξ: P = ξ ζ = τ T O ω + f T v O = 0

18 Statics of Rigid Bodies 18/72 Reciprocal wrenches and twists: A couple and a translation: ζ = [ ] [ 0 0, ξ = τ v] ζ ξ = 0 A force perpendicular to a translation: ζ = [ ] f, ξ = 0 [ ] 0 v ζ ξ = f T v = 0

19 Statics of Rigid Bodies 19/72 Reciprocal wrenches and twists: A couple perpendicular to a rotation: ζ = [ ] [ 0 ω, ξ = τ 0] ζ ξ = τ T ω = 0 A force intersecting or parallel to a rotation: [ [ ] [ ] f f ζ 1 =, ζ 0] 2 = 2 e = f r 2 f 2 2 r 2 e ξ = ζ 1 ξ = 0 [ ] ω = ω 0 [ ] e 0 ζ 2 ξ = ωf 2 e (r 2 e) = 0

20 Freedoms and Constraints 20/72 Wrench space of a serial chain A wrench ζ that is reciprocal to all joint twists of a serial chain, ξ 1,, ξ µ, is reciprocal to any possible motion of the EE and, thus, cannot exert any work on it: P = ζ ξ EE = ζ µ = ζ ξ i = 0 i=1 µ i=1 ξ i ζ may be interpreted as a constraint or reaction wrench, namely a generalized force acting on the EE that requires no motor actions to be balanced.

21 Freedoms and Constraints 21/72 The set of all wrenches reciprocal to ξ EE form the end-effector constraint wrench space W. Any ζ W satisfies the relationship: ˆζ T ξ 1 = = ˆζ T ξ µ = 0 ˆζ T [ ] ξ 1 ξ 2 ξ µ = ˆζT J = 0 Hence W is the (left) null space of the Jacobian matrix J, namely is the orthogonal complement (in ray coordinates) of T (which is the column space of J): W = T Clearly: dim W = dim se(3) dim T = 6 rank(j) Also: T = W The only possible motions of the EE are those reciprocal to W!

22 Freedoms and Constraints 22/72 Twist and wrench space of a parallel chain Twist space of the j th leg: T j = span(ξ 1j,, ξ µj ) The twist space of the EE is the intersection of the legs T j : T = N j=1 T j Constraint space imposed by the j th leg on the EE: W j = T j The overall constraint space on the EE is the sum (linear span) of the legs W j : W = N j=1 W j = T

23 Freedoms and Constraints 23/72 Example W = span {ζ }

24 Freedoms and Constraints 24/72 Example W = span { ζ 0}

25 Freedoms and Constraints 25/72 Example W = span {ζ 1, ζ 2 }

26 Freedoms and Constraints 26/72 Example W j = span ( ζj1, ζj2 ), with ζ j1 and ζj2 couples to e j W = W 1 +W 2 +W 3 =span(ζ11,ζ 12,ζ 21,ζ 22,ζ 31,ζ 32)=span (ζ1,ζ2,ζ3 ) T = W = span (ξ1, ξ2, ξ3 ) The end-effector may perform 3-dimensional translations.

27 Screw Systems 27/72 Motivation The twist space T of a serial chain is generated by the linear span of the joint twists: ξ EE T = span {ξ 1,, ξ µ } = span {σ 1,, σ µ } Having a picture of T is important, for several reasons: to determine the location and pitch of the end-effector twist; to determine whether the joint screws of a chain are at a singular (linearly dependent) configuration; to determine whether a chain can realize specific motions along certain directions (e.g. pure rotations or pure translations).

28 Screw Systems 28/72 If ξ T and λ R, then λξ T. Accordingly, T may simply be described by the screws underlying the twists. The result is what is called a screw system.

29 Screw Systems 29/72 Hunt s classification of screw systems (1978) There are: 6 types of 2-systems general 2-system 5 special 2-systems 11 types of 3-systems general 3-system 10 special 3-systems 6 types of 4-systems general 4-system 5 special 4-systems 2 types of 5-systems general 5-system special 5-system We will focus on some particular interesting cases.

30 2-Systems 30/72 Special 2-systems: The 1 st special 2-system is a pencil of equal-pitch screws lying on a plane and converging in a point O. ξ α and ξ β form a basis of T : T = span {ξ α (0), ξ β (0)}. The 2 nd special 2-system is a pencil of parallel equal-pitch screws lying on a plane.

31 2-Systems 31/72 Special 2-systems: The 3 rd special 2-system is a pencil of -pitch screws perpendicular to a given direction n. The 5 th special 2-system comprises screws of all pitches on a given line.

32 Special 3-systems 32/72 2 nd special 3-system with 0 pitch This system defines the freedoms of the instantaneous spherical motion, which comprises all possible rotations about a fixed point:

33 Special 3-systems 33/72 4 th special 3-system with 0 pitch The 4 th special 3-system comprises: screws of pitch 0 on all lines of a plane Γ; an -pitch screw perpendicular to Γ. The 3-dof serial arm (with all revolute axes lying on plane Γ in the shown configuration) may perform instantaneous pure rotations about any line of Γ, as well as a pure translation perpendicular to Γ. This screw system plays an important role in the design of homokinetic (constant-velocity) couplings and parallel manipulators with 2 rotational and 1 translational dofs.

34 Special 3-systems 34/72 5 th special 3-system with 0 pitch This system defines the freedoms of the instantaneous planar motion, comprising all possible rotations about lines parallel to a given direction and all translations perpendicular to it.

35 Special 3-systems 35/72 6 th special 3-system The 6 th special 3-system comprises -pitch screws along all directions in space (3D translational motion).

36 4-systems 36/72 The 3 rd special 4-system The 3 rd special 4-system (or Schönflies system) comprises screws of all pitches along all lines parallel to direction e, with -pitch screws in all directions in space.

37 5-systems 37/72 Special 5-system The special 5-system comprises: -pitch screws along all directions in space; finite-pitch (including 0) screws on all lines perpendicular to a given direction e.

38 Invariant Screw Systems (ISSs) 38/72 Example: a chain with 3 convergent R joints T = span(ξ 1 (0), ξ 2 (0), ξ 3 (0)), dim T = 3. Under arbitrary (nonsingular) displacements of the chain L, T comprises a bundle of 0-pitch screws through O (2 nd special 3-system). Therefore, T is invariant.

39 Invariant Screw Systems (ISSs) 39/72 Example: a chain with 4 convergent R joints T = span(ξ 1 (0), ξ 2 (0), ξ 3 (0), ξ 4 (0)), dim T = 3, 2 nd special 3-system. Under arbitrary (nonsingular) displacements of the chain L, T comprises a bundle of 0-pitch screws through O. The screws of L remain linearly dependent for finite displacements of L. T is invariant.

40 Invariant Screw Systems (ISSs) 40/72 Example: a chain with 4 parallel R joints T = span(ξ 1 (0), ξ 2 (0), ξ 3 (0), ξ 4 (0)), dim T = 3, 5 th special 3-system. Under arbitrary (nonsingular) displacements of L, T comprises a bundle of 0-pitch screws parallel to e and all -pitch screws perpendicular to e. The screws of L remain linearly dependent for finite displacements of L. T is invariant.

41 Invariant Screw Systems (ISSs) 41/72 Example: a 4-dof Schönflies chain T = span(ξ 1 (0), ξ 2 (0), ξ 3 (0), ξ 4 ( )), dim T = 4, 3 th special 4-system. The 3 rd special 4-system defined by the serial chain L does not change as the chain moves, since direction e is constant. The screw system is invariant.

42 Invariant Screw Systems (ISSs) 42/72 Invariant screw systems (ISSs): Whichever finite displacement is assigned around the joint screws of L (besides a discrete number of possibly singular configurations), the screw system instantaneously generated is invariant, thus preserving: dimension, type, shape, pose.

43 Invariant Screw Systems (ISSs) 43/72 ISSs guarantee full-cycle mobility It is generally not possible to infer information about the finite (or full-cycle) motion of a chain from the instantaneous motion in one configuration. Instead, this is possible if the instantaneous motion is described by an ISS. ISSs are the subalgebras of the Lie algebra se(3) of the special Euclidean group SE(3). The subalgebras are the tangent spaces at the identity of the (connected) subgroups of SE(3). If the instantaneous motion of the end-effector of a serial chain is described by a subalgebra, the finite motion lies within the corresponding subgroup.

44 Invariant Screw Systems (ISSs) 44/72 Example The end-effector may perform: instantaneous rotations about any line parallel to e instantaneous translations along any direction perpendicular to e finite rotations about any line parallel to e finite translations along any direction perpendicular to e

45 Invariant Screw Systems (ISSs) 45/72 Full-cycle linear dependence of the joint screws of an ISS chain ξ 1, ξ 2, ξ 3, ξ 4 are all parallel to e; T = span(ξ 1, ξ 2, ξ 3, ξ 4 ) is a planar-motion ISS (dim T = 3). For generic arbitrary motions of the chain, ξ 1, ξ 2, ξ 3 and ξ 4 remain parallel, thus forming a planar-motion ISS. T does not change even if a 5 th hinge parallel to e is added to the chain: T = span(ξ 1,..., ξ 5 ) dim T = 3

46 Invariant Screw Systems (ISSs) 46/72 Counter-example Singular configuration: ξ 1, ξ 2, ξ 3 and ξ 4 are all coplanar. T = span(ξ 1, ξ 2, ξ 3, ξ 4 ) is a 4 th special 3-system (dim T = 3). ξ 1, ξ 2, ξ 3 and ξ 4 are instantaneously linearly dependent. Out of the singular configuration: ξ 1, ξ 2, ξ 3 and ξ 4 are not coplanar. dim T = 4.

47 Persistent Screw Systems (PSSs) 47/72 Example: a persistent chain T = span(ξ 1,..., ξ 6 ), dim T = 5, 1 st special 5-system. T comprises all screws reciprocal to ζ( ), which is an -pitch wrench parallel to n = e 1 e 2. As the chain L displaces, n and ζ( ) change orientation. T preserves dimension, shape but not its pose. In every nonsingular configuration, L generates a persistent screw system (PSS).

48 Application: Mobility Analysis 48/72 In general, if the chain generates an ISS of dimension d, the finite mobility of the closed-loop mechanism obtained by blocking the end-effector is: M = i fi d where f i is the number of freedoms of the generic i th joint. Full-cycle linear dependence of joint screws allows closed-loop mechanisms with predictable mobility to be easily built. Example: The chain has 5 freedoms, θ 1,, θ 5, but the EE has only 3 dof. If the end-effector is blocked (i.e. fixed to the frame), the mobility of the resulting closed-loop mechanism is: M = 5 3 = 2.

49 Application: Mobility Analysis 49/72 Example: Sarrus 6-bar linkage The persistent chain has 6 freedoms, but the EE has only 5 dof. If the EE is blocked, the mobility of the resulting closed-loop mechanism is: M = 6 5 = 1

50 Application: Mobility Analysis 50/72 Example: a 1-dof 6R linkage The persistent 6-R chain spans a 5-dimensional twist space and exerts a constraint force through O for full-cycle motions. If the end-effector is blocked, the mobility of the resulting closed-loop mechanism is: M = 6 5 = 1

51 Application: Mobility Analysis 51/72 Example: a spherical 4-bar linkage Ordinarily, no 3 joint axes in a spherical 4-bar linkage are coplanar. In the shown configuration, 3 R-pairs are coplanar, thus forming a 1 st special 2-system. The mobility of this subchain is 1: i f i = 3, d = 2, M sub = 3 2 = 1 However, for the overall chain: i f i = 4, d = 3, M = 4 3 = 1 The R-joint ( ) is transitorily inactive, thus having zero instantaneous angular velocity: stationary configuration.

52 Application: Mobility Analysis 52/72 Example: a planar 4-bar linkage Ordinarily, no 3 joint axes of the planar 4-bar linkage are coplanar. In the shown configuration, 3 R-pairs are coplanar, thus forming a 2 nd special 2-system. The mobility of this subchain is 1: i f i = 3, d = 2, M sub = 3 2 = 1 However, for the overall chain: i f i = 4, d = 3, M = 4 3 = 1 The R-joint ( ) is transitorily inactive, thus having zero instantaneous angular velocity: stationary configuration.

53 Application: Design 53/72 Synthesis of parallel manipulators T, the twist space of the EE, is the space of all possible instantaneous motions of the EE: T = n i=1 T i T, the wrench space of the EE, is the space of the constraint wrenches that allow motions in T only: T = n i=1 T i

54 Application: Design 54/72 Synthesis procedure: assign T ; determine T (it is a linear operation); choose a suitable generating set for T : T = span(ζ 1,..., ζ m ), with m dim(t ) assign the generating wrenches of T to legs: { ζ11, ζ 12 leg 1 m = 4 ζ 21, ζ 22 leg 2 { T 1 = span(ζ 11, ζ 12 ) for each Ti, determine T i (it is a linear operation); T 2 = span(ζ 21, ζ 22 )

55 Application: Design 55/72 for each T i, design a leg so that its joint screws belong to T i : T i = span(ξ i1, ξ i2,..., ξ iµ ) The design process is very simple, but... purely local! Nothing guarantees that the desired motion characteristics hold in other configurations! If T i is an ISS: T i is invariant i T i is invariant T is invariant

56 Application: Design 56/72 Design of CV transmissions Constant-velocity (CV) transmissions between intersecting shafts are designed so as to guarantee: 2-dof of relative rotation; 1-dof of relative translation; angular-velocity transmission with unitary transmission ratio ω in = ω out

57 Application: Design 57/72 Screw theory proves that a CV transmission is reached as long as the relative connection between the shafts is built so that it generates a 4 th special 3-system with 0 pitch:

58 Kinematics Statics Freedoms Screw Systems Application: Design ISS and PSS Mobility Design Singularities 58/72 The classic Rzeppa joint, for instance, is kinematically equivalent to a parallel mechanism with µ RSR architecture (µ 3):

59 Application: Design 59/72 Each RSR leg is mirror symmetric about plane Π (called homokinetic or bisecting plane) and generates a constraint force on Π. If µ 3 and the constraint forces are linearly independent (they in general do not pass through the same point), a 4 th special 3-system is obtained on Π.

60 Application: Singularity Analysis 60/72 Singularity analysis of a serial chain: The chain generates a 5-system T with a single constraint force: T = span {ξ 1,, ξ 5 } T = span {ζ 1 (0)} When ξ 2, ξ 3, ξ 4 become coplanar (thus forming a 2 nd special 2-system), T degenerates into a 4-system, whose constraint space T is spanned by two 0-pitch wrenches: T = span {ζ 1 (0), ζ 2 (0)}

61 Application: Singularity Analysis 61/72 Singularity analysis of a parallel CV-coupling: The EE constraint wrench space T is spanned by a bundle of forces through O and a force ζ 4 (0) through P : T = span {ζ 1 (0), ζ 2 (0), ζ 3 (0), ζ 4 (0)} T contains a constraint torque ζ 4( ) perpendicular to plane π.

62 Application: Singularity Analysis 62/72 In the in-line configuration: the constraint space of the 5R leg is bidimensional, and it is spanned by a torque ζ 4( ) perpendicular to π and a force ζ 5(0) through O; T remains unchanged with respect to a full-cycle condition, while a transitory internal freedom appears within the subchain formed by twists ξ 2(0), ξ 3(0) and ξ 4(0), which belong to the same 2 nd special 2-system.

63 Kinematics Statics Freedoms Screw Systems ISS and PSS Mobility Application: Singularity Analysis Design Singularities 63/72 Singularity analysis of the 3-UPU parallel manipulator Each UPU leg generates a 5-system T with a -pitch constraint wrench Ti = span {ζi ( )} T = span {ζ1 ( ),, ζ3 ( )} T is a 6th special 3-system (translational motion).

64 Kinematics Statics Freedoms Screw Systems ISS and PSS Mobility Design Application: Singularity Analysis Singularities 64/72 When the constraint wrenches ζi s become coplanar, T degenerates into a 3rd special 2-system and T becomes a 4-system. A passive constraint singularity (PCS) occurs: the 3 actuators of the robot cannot control the EE, which is instantaneously labile.

65 Application: Singularity Analysis 65/72 By actuating the prismatic joints, an active constraint is generated by each leg, namely an actuation force. In general, the 3 passive and the 3 active constraints acting on the EE span a 6-dimensional wrench space, meaning that the EE is completely controlled.

66 Application: Singularity Analysis 66/72 In this case, the actuation forces ζ 4 (0), ζ 5 (0) and ζ 5 (0) are coplanar and pass through the same point, so that: dim (span {ζ 1 ( ), ζ 2 ( ), ζ 3 ( ), ζ 4 (0)ζ 5 (0), ζ 6 (0)}) = 5 < 6 An active constraint singularity (ACS) occurs: the actuators cannot fully control the EE.

67 Application: Singularity Analysis 67/72 Singularity analysis of the DELTA robot: Each leg exerts between the base and EE a constraint torque orthogonal to the parallelogram plane: T = span {ζ 1 ( ), ζ 2 ( ), ζ 3 ( )}

68 Application: Singularity Analysis 68/72 When the torques are linearly dependent, e.g. they are perpendicular to the same direction n, the dimension of T increases to 4 and the mechanism is at a passive constraint singularity (PCS): dim (span {ζ 1 ( ), ζ 2 ( ), ζ 3 ( )}) = 2 < 3

69 Application: Singularity Analysis 69/72 However, if a simultaneous leg singularity occurs (all R and S joints in one leg become coplanar), a further constraint wrench ζ 4 (0) is exerted. The EE preserves its mobility, i.e. dim T = 3, but with a qualitative change, since the EE may now also rotate.

70 Application: Singularity Analysis 70/72 After actuating the base-mounted R joints: the actuator in the singular leg exerts no additional constraint between the base and the platform; the non-singular legs generate actuation forces ζ 5(0) and ζ 6(0), forming a 2 nd special 2-system comprising a torque perpendicular to n; the dimension of the overall (passive and active) constraint space is 4, thus leaving the EE with 2 uncontrollable local freedoms: dim(t + span{ζ 5, ζ 6}) = 4

71 Conclusions 71/72 Screws are geometrical entities that represent both the instantaneous motion of a rigid body (twist) and the set of generalized forces acting upon it (wrench). Screw theory naturally provides the geometrical and algebraic concepts and tools underlying the first-order kinematics and statics of rigid bodies. Screw theory provides a strong geometrical insight into many complex physical phenomena that engineers have to deal with, such as: mobility analysis, singularities, constraint design, type synthesis of mechanisms.

72 End of presentation 72/72 Presentation by: Marco Carricato, Yuanqing Wu GRAB Group of Robotics, Automation and Biomechanics University of Bologna Italy