1 chapter 1 review mobile robot kinematics

SYSTEM DYNAMICS & CONTROL CHAPTER REVIEW MOBILE ROBOT KINEMATICS & LOCOMOTION Dr Vo Tuong Quan HCMUT - 2011 Class Rules Class attendance: 10% (if a student absent more than time, the attendance score is zero) Homeworks & Class exercises report: 30% (students will submit the homework via ELearning system) Teamwork project : 20% • Each team includes 3-4 students, each team will present their research topic at the final weeks of the semester • English power point and english presentation in class are encouraged Final exam: 40% (Students will allow to use their documents while doing the exam Time for final exam: 90 minutes  2011 – Vo Tuong Quan Mobile Robot Kinematics Kinematics problems Kinematic model for a mobile robot  Consider the robot as a rigid body on wheels, operating on a horizontal plane How many dimenssion of this kinematic model of mobile robot? The total dimensionality of this kinematic model on the plane is three: - Two for position in the plane - One for orientation along the vertical axis, which is orthogonal to the plane  To specify the position of the robot on the plane we establish a relationship between the global reference frame (global coordinate) of the plane and the local reference frame (local coordinate) of the robot  2011 – Vo Tuong Quan Mobile Robot Kinematics YI The axes XI and YI define an arbitrary initial basis on the plane as the global reference frame from some origin O: {XI,YI} YR XR  P XI  2011 – Vo Tuong Quan Mobile Robot Kinematics YI Define position of the robot  choose a point P on the robot chassis as its position reference point The basis {XR, YR} defines two axes relative to P on the robots chassis YR XR  P XI  2011 – Vo Tuong Quan Mobile Robot Kinematics YI YR XR  P The position of P in the global reference frame is specified by coordinates x and y, and the angular difference between the global and local reference frame is given by  The pose of the robot is a vector with three elements: I = [ x y  ] XI  2011 – Vo Tuong Quan Mobile Robot Kinematics YI The orthogonal matrix: YR XR  P  cos  R    sin   sin  cos  rotation 0 0 1 This matrix can be used to map motion in the global reference frame {XI, YI} to motion in terms of the local reference frame {XR, YR} XI  2011 – Vo Tuong Quan Mobile Robot Kinematics This operation is denoted by R( )I because the computation of this operation depends on the value of  : R = R(/2) I  cos  R     sin   sin  cos  0 0 1  0   R    0 2  0   2011 – Vo Tuong Quan Mobile Robot Kinematics Given some velocity (x’, y’, ’) in the global reference frame  We can compute the components of motion along this robot’s local axes XR and YR In this case, due to the specific angle of the robot, the motion along XR is equal to y’ and motion along YR is -x’  0  x   y          R  R  I   0  y    x  2  0 1       2011 – Vo Tuong Quan Mobile Robot Kinematics Robot within an arbitrary initial frame • Initial frame: {XI, YI} x • Robot frame: {XR, YR} • Robot position: y YI YR I XR  • Mapping between the two frames:  x  R  R  I  R     y     2011 – Vo Tuong Quan P XI  cos  R    sin   sin  cos  0 0 1 10 Mobile Robot Kinematics  T Example: Establish the robot speed   x y  as a function of the wheel speeds φi, steering angles βi and steering speeds βi and the geometric parameters of the robot (configuration coordinates) yI • Forward kinematics v(t) s(t)  x     y   f ( 1 , n , 1 ,  m , 1 ,  m )    • Inverse kinematics     n 1   m xI   m  T  f ( x, y ,) 11  2011 – Vo Tuong Quan Mobile Robot Kinematics The forward kinematic model of a differential drive robot is relatively straight-forward The differential drive robot has two wheels each of diameter r Given a point P centred between the two drive wheels, each wheel is a distance l from P Given r, l, θ and the spinning speed of each wheel, φ1 and φ2 a forward kinematic model would predict the robots overall speed in the global reference frame as:  x  I   y   f l , r ,  , 1 ,     12  2011 – Vo Tuong Quan Mobile Robot Kinematics Compute a robot’s motion in the global reference frame from the motion in its local reference frame I = R(θ)-1R  The strategy is to first compute the contribution of each of the two wheels in the local reference frame and then convert these to the global reference frame The contribution of each wheel’s spinning speed to the translation speed at P in the direction of +XR is given as: r1 x r1  r xr  The total translation speed at P is given as: x R  x r1  x r The speed y R is even easier at it must always be zero 13  2011 – Vo Tuong Quan Mobile Robot Kinematics The contribution of each wheel’s spinning speed to the rotation speed at P is given as: r  r 1  2l 2  The total rotation speed at P is given as: 2l R  1  2  Kinematic model for a differential drive robot as:  r1 r     cos   sin  0 1  1   I  R   R     sin  cos  0  r  r   0   1  2l   2l 14  2011 – Vo Tuong Quan Mobile Robot Kinematics Suppose the robot is positioned such that θ = π/2, r = 1, and l = and the robot engages its wheels unevenly with φ1 = and φ2 =  The velocity in the global reference frame as:  x  0  0 3 0           I   y   1 0 0  3   0 1 1 1 15  2011 – Vo Tuong Quan Mobile Robot Kinematics Do we need to consider about the dynamics of mobile robot? 16  2011 – Vo Tuong Quan Mobile Robot Locomotions Wheels based mobile robot: - Wheels are the most appropriate solution for most robotic applications - There are lots of options for combinations of different kinds of wheels in different orientations - Selection of wheels depends on the application Three key considerations when designing wheel robots: - Stability - Maneuverability - Controllability 17  2011 – Vo Tuong Quan Mobile Robot Locomotions Wheels types a) Standard wheel: Two degrees of freedom; rotation around the (motorized) wheel axle and the contact point b) Castor wheel: Three degrees of freedom; rotation around the wheel axle, the contact point and the castor axle a) b) 18  2011 – Vo Tuong Quan Mobile Robot Locomotions c) Swedish wheel: Three degrees of freedom; rotation around the (motorized) wheel axle, around the rollers and around the contact point d) Ball or spherical wheel: Suspension technically not solved c) d) swedish 90° s wedis h 45° 19  2011 – Vo Tuong Quan Mobile Robot Locomotions Wheels arrangements • Legend Steered standard wheel Connected wheels Motorised Swedish wheel Motorised standard wheel Un-powered standard wheel Un-powered omnidirectional wheel • Two wheels 20  2011 – Vo Tuong Quan Mobile Robot Locomotions • Three wheels 21  2011 – Vo Tuong Quan Mobile Robot Locomotions • Four wheels 22  2011 – Vo Tuong Quan Mobile Robot Locomotions • Six wheels 23  2011 – Vo Tuong Quan ...  1 0 0  3   0 1   1   1  15  2 011 – Vo Tuong Quan Mobile Robot Kinematics Do we need to consider about the dynamics of mobile robot? 16  2 011 – Vo Tuong Quan Mobile Robot. .. wheel Un-powered omnidirectional wheel • Two wheels 20  2 011 – Vo Tuong Quan Mobile Robot Locomotions • Three wheels 21  2 011 – Vo Tuong Quan Mobile Robot Locomotions • Four wheels 22  2 011 –...   1  2l   2l 14  2 011 – Vo Tuong Quan Mobile Robot Kinematics Suppose the robot is positioned such that θ = π/2, r = 1, and l = and the robot engages its wheels unevenly with 1 = and