L ECTURE 6: D EGREES OF M ANEUVERABILITY I NSTANTANEOUS C ENTER OF R - PowerPoint PPT Presentation

16-311-Q I NTRODUCTION TO R OBOTICS L ECTURE 6: D EGREES OF M ANEUVERABILITY I NSTANTANEOUS C ENTER OF R OTATION I NSTRUCTOR : G IANNI A. D I C ARO

A R O B O T W I T H N S TA N D A R D W H E E L S O U T O F M W H E E L S • Rolling constraints: Rolling constraints J 1 ( β s ) R ( θ ) ˙ ξ I − J 2 ˙ ϕ = 0 " # " # ϕ f ( t ) J 1 f ϕ ( t ) = J 1 ( β s ) = J 2 = diag ( r 1 , r 2 , . . . , r N ) , , ϕ s ( t ) J 1 s ( β s ) ϕ ( t ) = ( N f + N s ) × 1 , J 1 = ( N f + N s ) × 3 J 1 is a matrix with projections for all wheels to their motions along their individual wheel planes, which have a fixed angle for the fixed wheels and a time-varying angle for the steerable wheels • Sliding constraints " # C 1 f C 1 ( β s ) R ( θ ) ˙ ξ I = 0 , C 1 ( β s ) = C 1 s ( β s ) C 1 = ( N f + N s ) × 3 C 1 is a matrix with projections for all wheels to their motions in the 2 direction orthogonal to wheels’ planes

B A C K T O T H E D E G R E E O F M O B I L I T Y 𝜺 𝙣 … • 𝜺 𝙣 quantifies the degrees of controllable freedom based on changes to wheels’ velocity, expressed by the equation system from the no lateral slip constraints applied to all standard wheels (fixed and steerable) • Each row of the projection matrix is a kinematic constraint : no-sliding, imposed by one of the wheels of the chassis along the direction orthogonal to the wheel where ξ R = R ( θ ) ˙ ˙ Projection matrix ξ I 2 3 x R ˙ h i P 1 f ⊥ P 1 f ⊥ P 1 f ⊥ 5 = 0 ˙ Standard fixed wheel 1: y R x R ˙ y R ˙ ˙ 4 θ R ˙ θ R Hold simultaneously 2 3 ˙ x R 4 5 h i P N f f ⊥ P N f f ⊥ P N f f ⊥ 5 = 0 Standard fixed wheel N f : y R ˙ x R y R 4 ˙ ˙ ˙ θ R ˙ θ R 2 3 x R ˙ h i 2 3 P 1 s ⊥ P 1 s ⊥ P 1 s ⊥ 5 = 0 Standard steering wheel 1: ˙ y R x R ˙ y R ˙ ˙ 4 θ R ˙ θ R 2 3 ˙ x R h i P N s s ⊥ P N s s ⊥ P N s s ⊥ 5 = 0 Standard steering wheel N s : ˙ y R x R ˙ y R ˙ ˙ 4 θ R ˙ 3 θ R

I N D E P E N D E N T # O F K I N E M AT I C C O N S T R A I N T S V S . M O B I L I T Y Is each one of these no-side motion constraint equations imposing an additional independent constraint to the robot kinematics? Independence → Are all constraint equations independent from each other? → " # " # J 2 ˙ ϕ J 1 f R ( θ ) ˙ Are the rows in the matrix (linearly) independent? ξ I = 0 C 1 f → rank[ C 1 ( 𝛾 s ) ] N ⨉ 3 → The greater the rank, the more constrained the motion is . • Mathematically, the constraint equations says that the vector 𝜊 must belong R to the null space N(C 1 ( 𝛾 s )) of the projection matrix C 1 ( 𝛾 s ) N ⨉ 3 N(C 1 ( 𝛾 s )) = { 𝒚 ∈ ℝ 2 ⨉ 𝕋 | C 1 ( 𝛾 s ) 𝒚 = 0 } • If matrix C 1 ( 𝛾 s ) has rank 3 (i.e., all 3 columns are independent), then the Null space only contains 0 (no motion) as possible solution to the homogeneous equation: no other motion vector can satisfy the constraints, robot can’t move! rank[ C 1 ( 𝛾 s ) ] N ⨉ 3 = 3 - dim N [C 1 ( 𝛾 s )] • The larger the Null space, the larger the set of motion vectors that can satisfy the no side motion constraints → the more motion freedom the robot has 4

D E G R E E O F M O B I L I T Y ( N O S I D E M O T I O N C O N S T R A I N T S ) Degree of mobility, 𝜺𝘯 𝜺𝘯 = dim N [C 1 ( 𝛾 s )] = 3 - rank[ C 1 ( 𝛾 s ) ] 0 ≤ rank[ C 1 ( 𝛾 s ) ] N ⨉ 3 ≤ 3 • no standard wheels (no side-motion constraints): rank[ C 1 ( 𝛾 s ) ] = 0 • all motion directions constrained: rank[ C 1 ( 𝛾 s ) ] = 3 • N [C 1 ( 𝛾 s )] ⊆ ℝ 2 ⨉ 𝕋 , the exact dimension of N [C 1 ( 𝛾 s )] is the same as the dimension of the vector basis B(N) that spans the null space of C 1 ( 𝛾 s ), which cannot be higher than 3, the dimension of ℝ 2 ⨉ 𝕋 • If dim[N] = 2 → The basis of N is made of two linearly independent vectors B(N) = ( v 1 , v 2 ) , which means that all feasible motion vectors v (that are in N) can be generated as a linear combination of these two vectors → One dimension of velocity is not directly controllable • If dim[N] =1 → Two dimensions of velocity are not directly controllable • if dim[N] =0 → All dimensions of velocity are not directly controllable • if dim[N] =3 → All dimensions of velocity are directly controllable 5

D I M E N S I O N O F N U L L S PA C E A N D C O N T R O L L A B L E D E G R E E S O F V E L O C I T Y Y R C 1 ( 𝛾 s ) → C 1f A L chassis rank[ C 1f ] 2 ⨉ 3 = 3 - dim N[ C 1f ] l P " # 0 1 0 (wheel AL ) C 1 f = l X R 0 1 0 (wheel AR ) A R N ( C 1 f ) = { ˙ ξ | C 1 f ˙ ξ R = 0 } 2 3 ˙ x R 8 x R ∈ R , free variable ˙ " # " # 0 1 0 0 ( r ˙ ϕ wheel AL ) > ( y R = 0 ˙ > < 6 7 ˙ 5 = ⇒ Solution set: y R = 0 ˙ y R 6 7 y R = 0 ˙ 0 1 0 0 4 > ( r ˙ ϕ wheel AR ) ˙ > θ R ∈ S , free variable ˙ : θ R All (feasible velocity) vectors in the null space are generated by the linear combination :             ˙ 1 0 1 0 x R        + ˙           ˙ 0 0 0 0 y R  = ˙ such that the basis B ( N ( C 1 f )) of N is: B = x R θ R  ,                    ˙   0 1 0 1 θ R   6

D I M E N S I O N O F N U L L S PA C E A N D C O N T R O L L A B L E D E G R E E S O F M O T I O N Y C 1 ( 𝛾 s ) → C 1f R rank[ C 1f ] 2 ⨉ 3 = 3 - dim N[ C 1f ] A L chassis " # l P 0 1 0 (wheel AL ) C 1 f = l X R 0 1 0 (wheel AR ) A R N ( C 1 f ) = { ˙ ξ | C 1 f ˙ ξ R = 0 }       1 0   • dim N[ C 1f ] = dim B( C 1f ) = 2         0 0 B ( N ( C 1 f )) =  ,     • rank[ C 1f ] = 3 - 2 = 1        0 1   . . • Directly controllable degrees of freedom in velocity: x and 𝝸 • Control in orientation is obtained by acting upon wheels’ speeds, no steering control is required 7

D E G R E E O F M O B I L I T Y: E X A M P L E S Di ff erential drive robot with two standard fixed wheels 1. Two wheels on the same axle: only one independent kinematic constraint (the second wheel is constrained by the axle) → rank(C1) = 1 → 𝜺𝘯 = 2 A. Control of the rate of change in orientation B. Control of forward/backward speed Tricycle chassis: one steerable + two fixed standard wheels 1. Two wheels on same axles + one on a di ff erent one: two independent kinematic constraints → rank(C1) = 2 → 𝜺𝘯 = 1 A. Control of forward/backward speed An explicit steering control is required to change orientation of front wheel, the two rear wheels get powered with same speed 8

D E G R E E O F M O B I L I T Y: E X A M P L E S Bicycle chassis: one fixed and one steerable wheel 1. Two wheels on di ff erent axles: two independent kinematic constraints → rank(C1) = 2 → 𝜺𝘯 = 1 A.Control of forward/backward speed An explicit steering control is required to change orientation (it can’t be done by velocity commands to the wheels) Pseudo-Bicycle chassis: two steerable wheels 1. Two steering wheels on di ff erent axles: two independent kinematic constraints → rank(C1) = 2 → 𝜺𝘯 = 1 A. Instantaneous control of forward/backward speed An explicit steering control is required to change orientation of both wheels (in a consistent manner) 9

D E G R E E O F M O B I L I T Y: E X A M P L E S Car-like Ackerman vehicle two standard fixed wheels + two steerable (but jointly constrained) wheels 1. Two rear wheels on the same axle + Two steering wheels connected by steering arms: four kinematic constraints, only two are linearly independent → C1 = C1f and rank(C1f) = 2 → 𝜺𝘯 = 1 A. Control of forward/backward speed An explicit steering control (onto the front wheels) is required to change orientation (it can’t be done by velocity commands to the wheels) Kinematically equivalent to bicycle model 10

D E G R E E O F M O B I L I T Y: E X A M P L E S Locked Bicycle: one fixed (steerable before) wheel + one fixed standard wheel 1. The front wheel in the same plane of rear wheel + locked in forward position: two independent kinematic constraints → rank(C1) = 2 → 𝜺𝘯 = 1 A. Control of forward/backward speed An explicit steering control is required to change orientation (it can’t be done by velocity commands to the wheels) ( l 1 = l 2 ) , ( β 1 = β 2 = π / 2) , ( α 1 = 0 , α 2 = π ) Y R If rank(C1f) > 1 ⇒ Motion is constrained on a line/circle Linearly independent! x R P → rank(C1f) = 2 " # " # cos( π / 2) sin( π / 2) l 1 sin( π / 2) 0 1 l 1 C 1 ( β s )) = C 1 f = = cos(3 π / 2) sin(3 π / 2) l 1 sin( π / 2) 0 − 1 l 1 11

E X A M P L E S F O R D E G R E E O F M O B I L I T Y Don’t focus on ICR (for now) … The degree of freedom of the robot motion Cannot move Fixed arc motion anywhere (No ICR) (Only one ICR) • Degree of mobility : 0 • Degree of mobility : 1 Fully free motion Variable arc motion ( ICR can be located (line of ICRs) at any position) • Degree of mobility : 2 • Degree of mobility : 3 12