MotionControlWMRsTT Slides

Autonomous and Mobile RoboticsProf. Giuseppe Oriolo

Motion Control of WMRs:Trajectory Tracking

Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 2

• feedback control is needed because the application of nominal (open-loop) control inputs would lead to unacceptable performance in practice

• kinematic models are used to design feedback laws because (1) dynamic terms can be canceled via feedback (2) wheel actuators are equipped with low-level PID loops that accept velocities as reference

• we consider a unicycle in the following

motion control


motion control problems

trajectory tracking(predictable transients)

posture regulation(no prior planning)


• the desired (reference) trajectory (xd(t), yd(t)) must be admissible: there must exist vd and !d such that

trajectory tracking

• thanks to flatness, given (xd(t), yd(t)) we can compute


• rather than using directly the state error qd — q , use its rotated version defined as

• we find

(e1, e2) is ep (previous figure) in a frame rotated by µ


• now use the input transformation

• we obtain

which is clearly invertible

lineartime-varying

nonlineartime-varying nonlinear


• linearize the error dynamics around the reference trajectory (e ¼ 0, hence sin e3 ¼ e3)

via approximate linearization

• define the linear feedback

• we obtain


• letting

• caveat: this does not guarantee asymptotic stability, unless vd and !d are constant (as on circles and lines); even in this case, asymptotic stability of the unicycle is not global (indirect Lyapunov method)

with a > 0, ³ 2 (0,1), the characteristic polynomial of A(t) becomes time-invariant

real negative

eigenvalue

pair of complexeigenvalues withnegative real part


• the actual velocity inputs v, ! are obtained plugging the feedbacks u1, u2 in the input transformation

• note: k2 ! 1 as vd ! 0, hence this controller can only be used with persistent cartesian trajectories (stops are not allowed)

• note:(v, !) ! (vd, !d) as e ! 0 (pure feedforward)

• global stability is guaranteed by a nonlinear version

if k1, k3 bounded, positive, with bounded derivatives


via input/output linearization

• choose the cartesian coordinates of point B as output

• idea: find an output whose dynamics can be made linear via feedback, i.e., with an input transformation


• if b 6= 0, we may set

obtaining

• differentiating wrt time

determinant = b


• µ is not controlled with this scheme, which is based on output error feedback (compare with the previous)

• the desired trajectory for B can be arbitrary; in particular, square corners may be included

• achieve global exponential convergence of y1, y2 to the desired trajectory letting

with k1, k2 > 0


simulations

n error

tracking a circle via approximate linearization


simulations

error

tracking an 8-figure via nonlinear feedback


simulations

tracking a square via i/o linearization

b = 0.75 ) the unicycle rounds the corners


simulations

tracking a square via i/o linearization

b = 0.2 ) accurate tracking but velocities increase

MotionControlWMRsTT Slides

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