16-311-Q I NTRODUCTION TO R OBOTICS L ECTURE 5 ( CONTINUED ): K INEMATIC CONSTRAINTS D EGREES OF FREEDOM / MOBILITY I NSTRUCTOR : G IANNI A. D I C ARO
M O B I L E R O B O T M A N E U V E R A B I L I T Y The kinematic mobility (maneuverability) of a robot chassis is its ability to directly move in the environment, which is the result of: 1. The rule that every standard wheel must satisfy its no sliding and rolling constraints ( ↔ Each wheel imposes zero or more constraints on the motion) � Determine the Degree of mobility, 𝜺 𝙣 : degrees of controllable freedom based on changes in wheels’ velocity 2. The additional freedom contributed by steering and spinning steerable wheels � Determine the Degree of steerability, 𝜺 𝘵 : degrees of controllable freedom based on steering wheels and then moving Mathematically, the degree of maneuverability 𝜺 M is defined as the sum of: Degree of mobility, 𝜺 𝙣 and Degree of steerability, 𝜺 𝘵 𝜺 M = 𝜺 𝙣 + 𝜺 𝘵 2
M A N E U V E R A B I L I T Y A N D ( N O N ) H O L O N O M Y The degree of mobility quantifies the controllable degrees of freedom of a mobile robot based on the changes applied to wheel velocities Holonomic : If the controllable degrees of freedom is equal to total degrees of freedom, then the robot is said to be Holonomic , Non-Holonomic otherwise 1. In the following we will first derive the controllable degrees of freedom using algebraic reasoning (matrix rank, notion of linear independence) 2. Then, we will show that kinematic constraints of a robot with respect to the degree of mobility can be also demonstrated geometrically using: Instantaneous center of rotation (ICR) / Instantaneous center of curvature (ICC) 3. Finally, we will relate maneuverability, DOFs, holonomic and non holonomic constraints, fully and under actuated robot systems 3
R E C A P : T Y P E S O F W H E E L S Fixed standard Steered standard / Orientable d Mecanum/Swedish Castor / Spherical O ff -centered orientable 4
F I X E D S TA N D A R D W H E E L Reference wheel point A (on the β : angle of wheel ˙ x axle) is in polar coordinates : A( l , α ) plane wrt chassis ξ R = R ( θ ) ˙ ˙ ˙ ξ I = R ( θ ) y ˙ θ Wheel axle e n a l p l e e Y R h W A ϕ v ˙ P X R • Rolling constraint (pure rolling at the contact point): All motion along the direction of the wheel plane is determined by wheel spin • Sliding constraint: The component of the wheel’s motion orthogonal to the wheel plane must be zero 5
F I X E D S TA N D A R D W H E E L : C O N S T R A I N T S E Q U AT I O N S The wheel, of radius r, spins over time such that its rotational position around the horizontal axle is a function of time: ˙ ϕ and linear velocity is r ˙ ϕ Rolling constraint: R [ ˙ θ ] projections of y ˙ x ˙ along wheel plane must equal linear velocity from wheel 3 component projection vectors Sliding constraint: 𝜷 , β , l are parameters R [ ˙ projections of θ ] y ˙ x ˙ orthogonal to the wheel in the local {R} frame plane must be zero 6
FIXED STANDARD WHEEL: ROLLING CONSTRAINT . . . . • R [ x y ] are the components of 𝜊 R the pose velocity 𝝸 vector in the coordinate frame {R} fixed to the robot in the reference point P. • Projections of all robot’s velocities (linear and angular) on the wheel’s velocity plane must equal the velocity implied by the wheel’s spinning (under pure rolling assumption) . • Projection of x along wheel velocity plane: . . x cos(90-( 𝜷 + 𝜸 )) → x sin( 𝜷 + 𝜸 ) . • Projection of y along wheel velocity plane: . y (-cos( 𝜷 + 𝜸 )) . • Projection of the robot angular velocity 𝛊 (- l ) 𝜷 + 𝜸 along wheel velocity plane: . 𝛊 (- l ) cos( 𝜸 ) 90-( 𝜷 + 𝜸 ) 7
FIXED STANDARD WHEEL: SUMMARY OF CONSTRAINTS (Pure) Rolling constraint 2 3 ˙ x R 5 = r ˙ ⇥ ⇤ ˙ Standard wheel A: P ˙ P ˙ P ˙ y R ϕ x R y R 4 θ R ˙ θ R y R , ˙ P ˙ x R , P ˙ y R , P ˙ θ R = Projections of ˙ x R , ˙ θ R along the wheel plane No sliding (no side motion) constraint 2 3 ˙ x R h i P ⊥ P ⊥ P ⊥ 5 = 0 ˙ Standard wheel A: y R x R y R ˙ ˙ ˙ 4 θ R ˙ θ R y R , ˙ P ⊥ x R , P ⊥ y R , P ⊥ θ R = Projections of ˙ x R , ˙ θ R orthogonal to the wheel plane ˙ ˙ ˙
N U M E R I C E X A M P L E : N O S I D E M O T I O N C O N S T R A I N T Wheel A is in position such that: α = 0, β = 0, θ = 0 Y Y I 2 1 0 0 3 2 3 2 3 ˙ ˙ x x R 0 1 0 6 7 6 7 6 7 ⇥ ⇤ ⇥ ⇤ 1 0 0 ˙ 5 = 1 0 0 ˙ 5 = 0 y y 6 7 6 7 6 7 4 5 4 4 0 0 1 ˙ ˙ θ θ v X R . No instantaneous motion is possible along A X I the x axis of the robot frame, dx/dt=0 Wheel plane • Projection of robot’s velocity component along the Y R axis on the wheel’s plane (perpendicular to v ) is 0, being perpendicular to each other • Projection of robot’s velocity component along the X R axis on the wheel’s plane is 1, being wheel’s plane and X R fully aligned • Projection of robot’s velocity component from rotation d 𝜄 /dt is 0, because of 𝛾 =0 • R ( 𝜄 ) = I because of no relative rotation between {R} and {I} 9
S TA N D A R D S T E E R A B L E W H E E L Additional degree of freedom: Now the angle β of the wheel wrt the chassis is not fixed anymore, but is time-varying , as result of the control actions Instantaneously, the rate of variation of β (t) does not have an impact on the motion constraints. It’s only by time integration that the changes in the steering angle have an impact on robot mobility. 10
C A S T O R W H E E L ✦ The rolling constraint is identical to the standard steering case since the o ff set axis does not play a role during motion ✦ The sliding constraint changes: the lateral force on the wheel occurs at A because this is the attachment of the wheel to the chassis ✦ All lateral motion is balanced by an equivalent caster steering motion such 11 that no resistance to motion is opposed by the castor wheel
C A S T O R W H E E L ✦ In a steered standard wheel, the steering action does not by itself cause a movement of the robot chassis. ✦ In a castor wheel the steering action itself moves the robot chassis because of the o ff set between the ground contact point and the vertical axis of rotation. Given any robot chassis motion ˙ ξ I , there exists some value for spin speed, ˙ ϕ , and steering speed, ˙ β , such that the constraints are met. Kinematic constraint equations are integrable! (steering must be free ) Therefore, a robot with only castor wheels can move with any velocity in the space of possible robot motions: omnidirectional system 12
S W E D I S H W H E E L ✦ A fixed standard wheel with rollers attached to the wheel perimeter, with axes that are antiparallel to the main axis of the fixed wheel component. ✦ The exact angle 𝜹 between the roller axes and the main axis can vary Capable of omnidirectional motion 13
S P H E R I C A L W H E E L ✦ Same equations as in the fixed case, but now β gives the direction of motion, and is a free variable derived from the sliding constraint equation. ✦ Example: If the robot translates in direction Y R, then the second equation becomes: sin( α + β ) = 0, making β = - α Omnidirectional motion. Neither rolling nor sliding constraints 14
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 A. Each wheel can impose zero or more constraints on the motion B. Only fixed and steerable standard wheels impose kinematic constraints Let’s assume to have N = Nf + Ns standard (Fixed + Steerable) wheels • 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 ) Sliding constraints • Lateral movement: " # C 1 f C 1 ( β s ) R ( θ ) ˙ ξ I = 0 , C 1 ( β s ) = C 1 s ( β s ) 15
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 16 direction orthogonal to wheels’ planes
D I F F E R E N T I A L D R I V E R O B O T E X A M P L E Y R A L chassis l P l X R A R l, Right wheel: A R ( α = - 𝛒 /2), β = 𝛒 l, Left wheel: A L ( α = 𝛒 /2), β = 0 No steering wheels " # " # " # " # J 1 ( β s ) J 2 ˙ ϕ J 2 ˙ ϕ J 1 f ➔ R ( θ ) ˙ R ( θ ) ˙ ξ I = ξ I = C 1 ( β s ) 0 0 C 1 f 17
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