Week 3: Wheeled Kinematics AMR - Autonomous Mobile Robots

Transcription

1 Week 3: Wheeled Kinematics AMR - Paul Furgale Margarita Chli, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Wheeled Kinematics 1

2 AMRx Flipped Classroom A Matlab exercise is coming later in the class. Download it now! Wheeled Kinematics 2

3 AMRx A review Please register in at edx.org and watch the lecture segments Please try the problem sets online (they don t count for your grade but they are useful) Bring your laptop along for the lectures We may provide short examples or problems for you to work out We may ask for feedback on the worked exercises Flipped Classroom You ask us, we try to help No recording of the lecture, no questions are stupid Feel free to interrupt Wheeled Kinematics 3

4 Why Wheeled Robots? + Requires small control action (in comparison to legs) + Energetically very efficient Wheeled Kinematics 4

5 Why Wheeled Robots? + Requires small control action + Energetically very efficient - Unable to overcome many obstacles Wheeled Kinematics 5

6 AMRx Today s lecture Teacher: Paul Furgale, Deputy Review Practical problems: Intuition about matrices Computing good steering angles Computing wheel odometry Wheeled Kinematics 6

7 Wheeled Kinematics Not all degrees of freedom of a wheel can be actuated or have encoders Wheels can impose differential constraints that complicate the computation of kinematics r φ Y W X W x y = φ r 0 rolling constraint no-sliding constraint Wheeled Locomotion Introduction 7

8 Differential Kinematics Differential forward kinematics Given a set of actuator speeds, determine the corresponding velocity Differential inverse kinematics Given a desired velocity, determine the corresponding actuator speeds Wheeled Locomotion Introduction 8

9 Deriving a general wheel equation S r SW W robot angular velocity steering rate r RS I r IR R v IW = v IR + ω IR r RS + ω IR r SW + ω RS r SW robot velocity steering offset wheel offset Wheeled Locomotion Differential Kinematics 9

10 Example: Standard wheel Rolling constraint Y R sin α + β cos α + β l cos β R θ ξi φ r = 0 Y W Y S X W β X S No-sliding constraint cos α + β sin α + β l sin β R θ ξi = 0 l α X R Wheeled Locomotion Differential Kinematics 10

11 Example: Standard wheel Y R This equation had a mistake in the online lecture. The slides have been updated and we will update the video soon Y W Y S X W β X S No-sliding constraint cos α + β sin α + β l sin β R θ ξi = 0 l α X R Wheeled Locomotion Differential Kinematics 11

12 Differential Kinematics Given a wheeled robot, each wheel imposes n constraints Only fixed and steerable standard wheels impose no-sliding constraints Suppose a robot has n wheels of radius r i, the individual wheel constraints can be concatenated in matrix form: Rolling constraints J 1 β S R θ ξi J 2 φ = 0, J 1 β S = No-sliding constraints C 1 β S R(θ)ξI = 0, C 1 β S = J 1f J 1S β S, J 2 = diag r 1,, r N, φ = C 1f C 1S β S φ 1 φ N Wheeled Locomotion Wheeled Kinematics 12

13 Differential Kinematics Stacking the rolling and no-sliding constraints gives an expression for the differential kinematics J 1 β s C 1 β s R θ ξ I = J 2 0 φ Solving this equation for ξi yields the forward differential kinematics equation needed for computing wheel odometry Solving this equation for φ yields the inverse differential kinematics equation needed for control Wheeled Locomotion Wheeled Kinematics 13

14 Degree of Maneuverability The Degree of Maneuverability, δ M, combines mobility and steerability δ M = δ m + δ s Examples Wheeled Locomotion Wheeled Kinematics 14

15 Worked Exercise A Differential Drive Robot Two fixed standard wheels The robot frame (R) in between the wheels Stack the wheel equations for this configuration b r φ l Y R X R r φ r Wheeled Kinematics 15

16 A Differential Drive Robot Summary Degree of Maneuverability δ m = 2, δ s = 0, δ M = 2 r φ l Forward differential kinematics x r/2 r/2 φ r y = 0 0 φ l θ r/2b r/2b b Y R Inverse differential kinematics φ r φ l = 1/r 0 b/r 1/r 0 b/r x y θ X R r φ r Wheeled Kinematics 16

17 An Omnidirectional Robot This robot was the example in the online problem set. Were there any questions about the example? Y R X R Wheeled Kinematics 17

18 Simple Robots r φ l Y R b Y R X R X R r φ r Wheeled Kinematics 18

19 Practical Problems Head over to MATLAB and get into groups Wheeled Kinematics 19

20 Computing the steering angle X W Y S v v x β v y To compute the steering angle, we must transform the velocity of the vehicle into the steering frame S, then determine the angle, β, that brings the velocity into the negative y direction in the wheel frame. β π 2 X S Y W β = π 2 + atan( v y/v x ) Wheeled Kinematics 20

21 Computing Wheel Odometry Given sensed wheel speeds at time t, we know how to generate a predicted platform velocity: Recall that ξr(t) = F φ (t) ξr = R θ ξi In the inertial frame, the velocity equation is: ξi(t) = R( θ(t))f φ (t) Wheeled Kinematics 21

22 Computing Wheel Odometry Assumptions: Timesteps of size T Constant acceleration ξr(t) = ξ R t 0 + t t 0 T ξr t 1 ξr t 0, t 0 t < t 1 The velocity integration equation: ξ I t k = ξ I t k 1 + t k ξi t k 1 t dt Wheeled Kinematics 22

23 Computing Wheel Odometry Assumptions: Timesteps of size T Constant acceleration ξr(t) = ξ R t 0 + t t 0 T ξr t 1 ξr t 0, t 0 t < t 1 The velocity integration equation: ξ I t k ξ I t k 1 + T 2 R( θ t k 1 )(ξr t k 1 + ξr(t k )) Wheeled Kinematics 23

24 Computing Wheel Odometry Assumptions: Timesteps of size T Constant acceleration ξr(t) = ξ R t 0 + t t 0 T ξr t 1 ξr t 0, t 0 t < t 1 The velocity integration equation: ξ I t k ξ I t k 1 + T 2 R( θ t k 1 )(F t k 1 φ (t k 1 ) + F(t k )φ (t k )) Wheeled Kinematics 24

25 Computing Wheel Odometry The velocity integration equation: ξ I t k ξ I t k 1 + T 2 R( θ t k 1 )(F t k 1 φ (t k 1 ) + F(t k )φ (t k )) Wheeled Kinematics 25