Kinematics of Mobile Robots Forward Kinematics of Differential
Kinematics of Mobile Robots
Forward Kinematics of Differential drive
Where can we find differential drives? 1. 2. 3. 4. 5. 6. Braitenberg Vehicles Micromouse Most of our robots Unicycle Segway Many many others
Kinematics of Differential drive Differential Drive is the most common kinematic choice All of the miniature robots… Pioneer, Rug warrior - difference in wheels’ speeds determines its turning angle Questions (forward kinematics) Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL VR Are there any inherent system constraints? 1) Specify system measurements 2) Determine the point (the radius) around which the robot is turning. 3) Determine the speed at which the robot is turning to obtain the robot velocity. 4) Integrate to find position.
Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y VL 2 d q VR (assume a wheel radius of 1) x
Kinematics of Differential drive – radius of turning 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. VL 2 d q x VR ICC (assume a wheel radius of 1) “instantaneous center of curvature”
Kinematics of Differential drive – angular velocity 1) Specify system measurements - consider possible coordinate systems y VL 2 d x q 2) Determine the point (the radius) around which the robot is turning. - to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles - each wheel must be traveling at the same angular velocity VR ICC (assume a wheel radius of 1) “instantaneous center of curvature” = angular velocity
Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y w VL 2 d x q 2) Determine the point (the radius) around which the robot is turning. - to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles - each wheel must be traveling at the same angular velocity around the ICC VR ICC (assume a wheel radius of 1) “instantaneous center of curvature”
Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d 3) Determine the robot’s speed around the ICC and its linear velocity VR ICC R robot’s turning radius (assume a wheel radius of 1) w(R+d) = VL w(R-d) = VR
Kinematics of Differential drive 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d VR ICC 3) Determine the robot’s speed around the ICC and then linear velocity “instantaneous center of curvature” ICC w(R+d) = VL R w(R-d) = VR robot’s turning radius Thus, (assume a wheel radius of 1) w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL )
Kinematics of Differential drive – robot’s velocity 1) Specify system measurements - consider possible coordinate systems y 2) Determine the point (the radius) around which the robot is turning. - each wheel must be traveling at the same angular velocity around the ICC w VL x 2 d 3) Determine the robot’s speed around the ICC and then linear velocity VR ICC w(R+d) = VL R w(R-d) = VR robot’s turning radius Thus, w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) So, the robot’s velocity is V = w. R = ( VR + VL ) / 2
Kinematics of Differential drive – integrate to obtain position 4) Integrate to obtain position y Vx = V(t) cos(q(t)) w(t) Vy = V(t) sin(q(t)) q(t) VL x 2 d VR ICC Vx “instantaneous center of curvature” ICC R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) What has to happen to change the ICC ? V = w. R = ( VR + VL ) / 2
Kinematics of Differential drive 4) Integrate to obtain position y Vx = V(t) cos(q(t)) w(t) Thus, VL x(t) = x 2 d Vy = V(t) sin(q(t)) y(t) = q(t) = VR ∫ V(t) cos(q(t)) dt ∫ V(t) sin(q(t)) dt ∫ w(t) dt ICC R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) V = w. R = ( VR + VL ) / 2
Kinematics of Differential drive – velocity components Velocity Components y Vx = V(t) cos(q(t)) speed Vy = V(t) sin(q(t)) w(t) Thus, VL x(t) = V(t) cos(q(t)) dt x 2 d y(t) = V(t) sin(q(t)) dt q(t) = VR w(t) dt ICC Kinematics R(t) robot’s turning radius with w = ( VR - VL ) / 2 d R = 2 d ( VR + VL ) / ( VR - VL ) What has to happen to change the ICC ? V = w. R = ( VR + VL ) / 2
Inverse Kinematics of Differential Drive
Inverse Kinematics – the problem Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position
Inverse Kinematics – one solution Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position
Inverse Kinematics – another solution Key question: Given a desired position or velocity, what can we do to achieve it? y x VL (t) VR(t) starting position final position
Inverse Kinematics – many numerical solutions to equations Key question: Given a desired position or velocity, what can we do to achieve it? y Need to solve these equations: x = V(t) cos(q(t)) dt y = V(t) sin(q(t)) dt x VL (t) q = w(t) dt w = ( VR - VL ) / 2 d V = w. R = ( VR + VL ) / 2 VR(t) starting position final position for VL (t) and VR(t). There are lots of solutions. . .
Inverse Kinematics – finding the best solution Key question: Given a desired position or velocity, what can we do to achieve it? y Finding some solution is not hard, but finding the “best” solution is very difficult. . . x VL (t) VR(t) starting position final position • quickest time • most energy efficient • smoothest velocity profiles VL (t) t It all depends on who gets to define “best”. . .
Inverse Kinematics - decomposition Usual approach: decompose the problem and control only a few DOF at a time Differential Drive y x VL (t) VR(t) starting position final position
Inverse Kinematics – decomposition for Differential Drive Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax x VL (t) VR(t) starting position final position
Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax (2) drive straight until the robot’s origin coincides with the destination x VL (t) = VR (t) = Vmax VR(t) starting position final position
Inverse Kinematics Usual approach: decompose the problem and control only a few DOF at a time Differential Drive (1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y -VL (t) = VR (t) = Vmax (2) drive straight until the robot’s origin coincides with the destination x VL (t) = VR (t) = Vmax VR(t) starting position (3) rotate again in order to achieve the desired final orientation final position -VL (t) = VR (t) = Vmax VL (t) VR (t) t
Problem Represent the forward and inverse kinematics for this robot using notation from the previous lecture.
Sources • • • Prof. Maja Mataric Dr. Fred Martin Bryce Tucker and former PSU students A. Ferworn, Prof. Gaurav Sukhatme, USC Robotics Research Laboratory Paul Hannah • Reuven Granot, Technion • Dodds, Harvey Mudd College
- Slides: 26