Autonomous and Mobile Robotics Prof. Giuseppe Oriolo Wheeled Mobile Robots 4 Motion Control of WMRs: Trajectory Tracking
motion control • a desired motion is assigned for the WMR, and the associated nominal inputs have been computed • to execute the desired motion, we need feedback control because the application of nominal inputs in open-loop would lead to very poor performance • dynamic models are generally used in robotics to compute commands at the generalized force level • kinematic models are used to design WMR 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 Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 2
• actual control scheme low-level PID loop reference velocity actual error motion commands motion high-level robot actuators PID control (dyn model) + + — — actual velocities (localization) • equivalent control scheme (for design) reference velocity actual error motion commands motion high-level robot control (kin model) + — (localization) Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 3
� � � � � � � � � � � � � � � � motion control problems posture regulation trajectory tracking (no prior planning) (predictable transients) • w.l.o.g. we consider a unicycle in the following Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 4
trajectory tracking: state error feedback • the unicycle must track a Cartesian desired trajectory ( x d ( t ), y d ( t )) that is admissible, i.e., there exist v d and ! d such that • thanks to flatness, from ( x d ( t ), y d ( t )) we can compute Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 5
• the desired state trajectory can be used to compute the state error, from which the feedback action is generated; whereas the nominal input can be used as a feedforward term • the resulting block scheme is reference input v d , ! d (feedforward) desired desired actual Cartesian state velocity state state trajectory trajectory trajectory commands trajectory via error unicycle flatness tracking p d q d v , ! + — q Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 6
• rather than using directly the state error q d — q , use its rotated version defined as ( e 1 , e 2 ) is e p (previous figure) in a frame rotated by µ • the error dynamics is nonlinear and time-varying Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 7
via approximate linearization • a simple approach for stabilizing the error dynamics is to use its linearization around the reference trajectory (indirect Lyapunov method ) local results) • to make the reference trajectory an unforced equilibrium for the error dynamics use the following (invertible) input transformation Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 8
• we obtain that is input term drift term nonlinear, linear in u nonlinear, time-varying Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 9
• hence, the linearization of the error dynamics around the reference trajectory is easily computed as • define the linear feedback • the closed-loop error dynamics is still time-varying! Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 10
• letting with a > 0, ³ 2 (0,1), the characteristic polynomial of A ( t ) becomes time-invariant and Hurwitz real pair of complex negative eigenvalues with eigenvalue negative real part • caveat: this does not guarantee asymptotic stability, unless v d and ! d are constant (rectilinear and circular trajectories); even in this case, asymptotic stability of the unicycle is not global (indirect Lyapunov method) Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 11
• the actual velocity inputs v , ! are obtained plugging the feedbacks u 1 , u 2 in the input transformation • note: ( v , ! ) ! ( v d , ! d ) as e ! 0 (pure feedforward) • note: k 2 ! 1 as v d ! 0, hence this controller can only be used with persistent Cartesian trajectories (stops are not allowed) • global stability is guaranteed by a nonlinear version if k 1 , k 3 bounded, positive, with bounded derivatives Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 12
• the final block scheme for trajectory tracking via state error feedback and approximate linearization is v d , ! d feedforward action p d q d e u v , ! via input rotation unicycle K flatness transf + — q feedback µ µ action • based on state error • needs v d , ! d • needs µ also for error rotation + input transformation Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 13
trajectory tracking: output error feedback • another approach: develop the feedback action from the output (Cartesian) error only, without computing a desired state trajectory, while the feedforward term is the velocity along the reference trajectory • the resulting block scheme is feedforward term actual desired velocity state Cartesian Cartesian trajectory commands trajectory error trajectory unicycle tracking v , ! p d + — p actual Cartesian trajectory Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 14
via exact input/output linearization • idea: (1) if the map between the available inputs and some derivative of the output is invertible, then (2) by inverting this map the system can be made linear • however, for the unicycle the map between the velocity inputs and the Cartesian output is singular as a consequence, input-output linearization is not possible in this case Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 15
� � � � � � � � � � • solution: change slightly the output so that the new input-output map is invertible and exact linearization becomes possible • displace the output from the contact point of the wheel to point B along the sagittal axis Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 16
• differentiating wrt time determinant = b • if b 6 =0 , we may set obtaining Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 17
• achieve global exponential convergence of y 1 , y 2 to the desired trajectory letting with k 1 , k 2 > 0 • 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 Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 18
• the final block scheme for trajectory tracking via output error feedback + input-output linearization is feedforward action e u v , ! q + unicycle p d + — + y µ feedback action • based on output error . • needs p d • needs x , y , µ for output reconstruction and µ also for input transformation Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 19
� ��� � � � � � � � �� �� �� ��� ��� ��� � � � � � �� �� � � simulations tracking a circle via approximate linearization �������� n error Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 20
� � � ��� � ��� � ��� � �� �� �� �� �� ��� ��� ��� ��� � � � � � � � � �� �� �� ��� simulations tracking an 8 -figure via nonlinear feedback ��������� error Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 21
�� � � ��� � ��� ��� ������� �������������������� � � � ��� � � � � � � � � � �� �� �� �� � � � � � �� �� �� ���� � ��� � ����� ������������������ � � � � � ��� simulations tracking a square via i/o linearization b =0.75 ) the unicycle rounds the corners Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 22
�� � � �� �� �� ��� ������� �������������������� � � � ��� � � � � � � � � � �� �� �� �� � � � � � �� �� �� ��� � ��� ��� ����� ������������������ � � � � � ��� simulations tracking a square via i/o linearization b =0.2 ) accurate tracking but velocities increase Oriolo: Autonomous and Mobile Robotics - Motion Control of WMRs: Trajectory Tracking 23
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