Tài liệu Kinematics

MOBILE ROBOTICS course KINEMATICS MODELS OF MOBILE ROBOTS Maria Isabel Ribeiro Pedro Lima [email protected] [email protected] Instituto Superior Técnico (IST) Instituto de Sistemas e Robótica (ISR) Av.Rovisco Pais, 1 1049-001 Lisboa PORTUGAL April.2002 All the rights reserved Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models References • Gregory Dudek, Michael Jenkin, “Computational Principles of Mobile Robotics”, Cambridge University Press, 2000 (Chapter 1). • Carlos Canudas de Wit, Bruno Siciliano, Georges Bastin (eds), “Theory of Robot Control”, Springer 1996. Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Kinematics for Mobile Robots • • • What is a kinematic model ? What is a dynamic model ? Which is the difference between kinematics and dynamics? • Locomotion is the process of causing an autonomous robot to move. – • Dynamics – the study of motion in which these forces are modeled – • In order to produce motion, forces must be applied to the vehicle Includes the energies and speeds associated with these motions Kinematics – study of the mathematics of motion withouth considering the forces that affect the motion. – – Robótica Móvel Deals with the geometric relationships that govern the system Deals with the relationship between control parameters and the beahvior of a system in state space. 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Notation Yb Ym Xm θ y P x • • Xb {Xm,Ym} – moving frame {Xb, Yb} – base frame x   q = y   θ robot posture in base frame  cos θ sin θ 0   R(θ) = − sin θ cos θ 0   0 1  0 Robótica Móvel Rotation matrix expressing the orientation of the base frame with respect to the moving frame 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Wheeled Mobile Robots • Idealized rolling wheel y axis x axis y axis z motion • If the wheel is free to rotate about its axis (x axis), the robot exhibits preferencial rollong motion in one direction (y axis) and a certain amount of lateral slip. • For low velocities, rolling is a reasonable wheel model. – This is the model that will be considered in the kinematics models of WMR Wheel parameters: • r = wheel radius • v = wheel linear velocity • w = wheel angular velocity Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Differential Drive ICC • 2 drive rolling wheels R L θ y ICC = ( x − R sin θ, y + Rcosθ) x • • • • vr(t) – linear velocity of right wheel control variables vl(t) – linear velocity of left wheel r – nominal radius of each wheel R – instantaneous curvature radius of the robot trajectory, relative to the mid-point axis R− w( t ) = v r (t) R+L 2 vl (t) R−L 2 Robótica Móvel Curvature radius of trajectory described by LEFT WHEEL R+ w(t ) = L 2 L 2 Curvature radius of trajectory described by RIGHT WHEEL w(t ) = R= v r (t) − v l (t) L L ( v l ( t ) + v r ( t )) 2 ( v l ( t ) − v r ( t )) 2002 - © Pedro Lima, M. Isabel Ribeiro v( t ) = w ( t )R = 1 ( v r ( t ) + v l ( t )) 2 Kinematics Models Differential Drive • Kinematic model in the robot frame v x ( t )  r 2 r 2      w l (t)  0   v y ( t ) =  0 !     w r ( t )   θ( t )  − r L r L   • • wr(t) – angular velocity of right wheel wl(t) – angular velocity of left wheel Useful for velocity control Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Differential Drive Kinematic model in the world frame 1 ( v r ( t ) + v l ( t )) 2 v (t) − v l (t) w(t ) = r L v( t ) = w( t )R = t ! x( t ) = v( t ) cos θ( t ) ! y( t ) = v( t ) sin θ( t ) ! θ( t ) = w( t ) x( t ) = ∫ v(σ) cos(θ(σ))dσ 0 t y( t ) = ∫ v(σ) sin(θ(σ))dσ 0 t θ( t ) = ∫ w(σ )dσ 0 !  x( t ) cos θ( t ) 0      v( t )  !   y( t ) =  sin θ( t ) 0  !     w( t )   1 θ( t )  0  Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro ! q( t ) = S(q)ξ( t ) control variables Kinematics Models Differential Drive • Particular cases: – vl(t)=vr(t) • Straight line trajectory v r ( t ) = v l ( t ) = v( t ) ! w( t ) = 0 ⇒ θ( t ) = 0 – ⇒ θ( t ) = cte. vl(t)=-vr(t) • Circular path with ICC (instantaneous center of curvature) on the mid-point between drive wheels v( t ) = 0 w( t ) = Robótica Móvel 2 vR (t) L 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Synchronous drive • In a synchronous drive robot (synchro drive) each wheel is capable of being driven and steered. • Typical configurations – – Three steered wheels arranged as vertices of an equilateral triangle often surmounted by a cylindrical platform All the wheels turn and drive in unison • • This leads to a holonomic behavior Steered wheel – The orientation of the rotation axis can be controlled y axis Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Synchronous drive • All the wheels turn in unison • All of the three wheels point in the same direction and turn at the same rate – This is typically achieved through the use of a complex collection of belts that physically link the wheels together • The vehicle controls the direction in which the wheels point and the rate at which they roll • Because all the wheels remain parallel the synchro drive always rotate about the center of the robot • The synchro drive robot has the ability to control the orientation θ of their pose diretly. • Control variables (independent) – v(t), w(t) t x( t ) = ∫ v(σ) cos(θ(σ))dσ 0 t y( t ) = ∫ v(σ) sin(θ(σ))dσ 0 t θ( t ) = ∫ w(σ )dσ 0 • The ICC is always at infinity • Changing the orientation of the wheels manipulates the direction of ICC Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Synchronous Drive • Particular cases: – v(t)=0, w(t)=w=cte. during a time interval • – The robot rotates in place by an amount v(t)=v, w(t)=0 during a time interval • ∆t w ∆t ∆t The robot moves in the direction its pointing a distance v ∆t Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Tricycle • • Three wheels and odometers on the two rear wheels Steering and power are provided through the front wheel • control variables: – – steering direction α(t) angular velocity of steering wheel ws(t) The ICC must lie on the line that passes through, and is perpendicular to, the fixed rear wheels ICC ICC R Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Tricycle Yb α d R θ y x Xb If the steering wheel is set to an angle α(t) from the straight-line direction, the tricycle will rotate with angular velocity w(t) about a point lying a distance R along the line perpendicular to and passing through the rear wheels. r = steering wheel radius v s (t) = w s (t) r ( linear velocity of steering wheel R( t ) = d tg π − α( t ) 2 w(t ) = ) w s (t) r d + R( t ) 2 2 w( t ) = Robótica Móvel angular velocity of the moving frame relative to the base frame v s (t) sin α( t ) d 2002 - © Pedro Lima, M. Isabel Ribeiro Kinematics Models Tricycle Kinematic model in the robot frame v x ( t ) = v s ( t ) cos α( t ) v y (t) = 0 with no splippage ! ( t ) = v s ( t ) sin α( t ) θ d Kinematic model in the world frame ! x( t ) = v s ( t ) cos α( t ) cos θ( t ) ! y( t ) = v s ( t ) cos α( t ) sin θ( t ) v (t) ! θ( t ) = s sin α( t ) d !  x( t ) cos θ( t ) 0      v( t )  !   y( t ) =  sin θ( t ) 0  !     w( t )   1 θ( t )  0 Robótica Móvel 2002 - © Pedro Lima, M. Isabel Ribeiro v( t ) = v s ( t ) cos α( t ) w(t ) = v s (t) sin α( t ) d Kinematics Models Omnidireccional 1 Ym L 30º Yf 2 θ 3 Xf Xm Kinematic model in the robot frame  0 Vx       2  Vy  = − r !   3 θ  r  3L  Robótica Móvel − 1 3 1 r 3 r 3L r 1  r 3   w1  1   r w 2  3   r  w 3  3L   2002 - © Pedro Lima, M. Isabel Ribeiro Swedish wheel w1, w2, w3 – angular velocities of the three swedish wheels Kinematics Models