- Email: [email protected]
Discussion on: “Trajectory Tracking for a Wheeled Mobile Robot Using a Vision Based Positioning System and an Attitude Observer”
360
Discussions on: “Observer Based Trajectory Tracking for a WMR”
Discussion on: “Trajectory Tracking for a Wheeled Mobile Robot Using a Vision Based Positioning System and an Attitude Observer” Masato Ishikawa Dept. of Mechanical Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
1. Quick Summary This paper by Velasco-Villa et al. [1] proposes an observerbased tracking control method for a wheeled planer mobile robot. The proposed method is simply summarized as follows.
from which the estimate for θ is uniquely reconstructed. Output feedback control is realized by replacing the unmeasurable information in the controller with the estimates.
2. Discussion
State Equation x˙ 1 = u1 cos θ,
θ˙ = u2 ,
x˙ 2 = u2 sin θ,
u˙ 1 = v1
where (x1 , x2 ) is the robot’s position, θ is the attitude angle and u1 is its velocity. The control input is U = (v1 , u2 )T where v1 is the acceleration and u2 is the angular velocity. A key assumption is that θ is not measurable. Tracking controller The state vector is split into y1 = (x1 , x2 )T and y2 = (θ, u1 )T . Let y1d and y2d be the respective reference trajectories, where y2d is dependent on y1d so that they satisfy the state equation, and y1e , y2e be the respective tracking error. The controller consists of a pair of P/PD feedback laws based on feedback linearization: −1 cos θ −u1 sin θ (¨y1d − Kd1 y˙ 1e − Kp1 y1e ) U1 = sin θ u1 cos θ −1 0 1 (˙y2d − Kp2 y˙ 2e ) U2 = 1 0 Choose U = U1 if |u1 (t)| ≥ δ, or U = U2 otherwise. Attitude observer
2.1. On Measurement Setting
η˙ 1 = −(η2 + x2 arctan u1 ) −u1 arctan u1 [η1 + arctan u1 ] −
x1 v1 1 + u12
η˙ 2 = −(η1 + x1 arctan u1 ) −u1 arctan u1 [η2 + arctan u1 ] −
x2 v1 1 + u12
where is the observer gain. η1 estimates (cos θ − x1 arctan u1 ) while η2 estimates (sin θ − x2 arctan u1 ), E-mail: [email protected]
Observer design and output feedback control for mobile robots is a challenging topic from both theoretical and practical points of view. A major difficulty is that the system is inherently nonlinear, thus the observer design and controller design is not simply separated as in the case of linear systems [2]. In particular, observability of the system is often spoiled when the robots remains stationary, or u1 = 0. This exemplifies a fundamental property of nonlinear systems that observability relies on control inputs [3], which makes things seriously awkward. With this respect, this paper proposes to combine 1) a switched controller to get out from the stationary (i.e., unobservable) states, with 2) an observer which is capable to estimate the attitude angle in spite of the spontaneous u1 . I read this paper with great interest. This paper provides a simple and practical example of observer design scheme based on the immersion and invariance (I&I) approach proposed by Astolfi et al. [4], followed by some physical experiments to prove its effectiveness. The I&I methodology allows us to consider indirect estimates η1 , η2 to obtain θ , which seems to be a promising approach for robust estimation. For all these contributions, I raise several points to be discussed.
First, I wonder which sort of realistic situation requires the problem setting, that the robot’s absolute position and velocity is measurable while the attitude angle is unmeasurable. Any example that supports this problem setting would be helpful. 2.2. On Convergence of the Estimation Error The authors assume that T := {t ∈ R|u1 (t) = 0} is a set of isolated time instants, namely, it does not contain nonempty interval nor accumulation point. Under
361
Discussions on: “Observer Based Trajectory Tracking for a WMR”
Fig. 2. Simulation under measurement noise with faster sampling.
• The tracking error y2e converges to zero as long as |u1 (t)| < δ holds forever. This is true. However, we should note that the two modes are switched to each other when |u1 (t)| cross the threshold δ, so the state may not remain in a single mode. Influence of switching should be taken into account to guarantee the whole convergence. This leads us to take closer look at the following cases. Fig. 1. Simulation under measurement noise with slower sampling.
this assumption, Proposition 1 claims convergence of the Lyapunov candidate V1 (t) to zero. Here the point we should take care is that V˙ 1 (t) is timedependent and vanishes when u1 (t) = 0. • Even if T is a finite set, V1 (t) does not necessarily converge to zero, because u1 (t) → 0 (t → ∞) may occur. I suggest this case should be excluded in advance by assumption. • For the case where T is infinite, the authors claim that S := V1 (T ) converges to zero because it is nonnegative and monotonically decreasing. In general, however, a nonnegative and monotonically decreasing series may converge to a nonzero limit. I suppose the conclusion itself can be justified by appropriately modifying the proof.
2.4. On Robustness and Possible Influence of Switching Fig. 1 shows a simulation result of the proposed scheme with ±5[mm] of measurement noise is imposed on x1 and x2 . The sampling period is set to 10[msec]. Other design parameters and the reference trajectory y1d = (0.0 sin t, 1 − 0.5 cos2 t)T are almost similar to the ones used in the paper. It seems to work quite nice in spite of the presence of measurement noise. However, if we shorten the sampling period to 1[msec], it starts to chatter around the threshold |u1 (t)| = δ as shown in Fig. 2. The chattering often occurs when the input v1 (t) takes difference signs in the both sides of the threshold, mainly because of the presence of measurement noise. We might overlook this phenomena when the sampling is relatively slow, but it is noteworthy that faster sampling (or faster switching) tend to make the system sensitive to noise.
2.3. On Convergence of the Tracking Error As for the stability of observer-controller combination, in Proposition 2, the authors proved that • The tracking error y1e converges to zero as long as |u1 (t)| ≥ δ holds forever.
Acknowledgment The author of this discussion is grateful to Prof. Teruyo Wada for her variable comments on Lyapunov stability and convergence analysis.
362
References 1. Velasco-Villa M, Aranda-Bricaire E, Rodrígues-Cotés H, González-Sierra J. Trajectory tracking for a wheeled mobile robot using a vision based positioning system and an attitude observer. Eur J Control, 2012; to appear.
Discussions on: “Observer Based Trajectory Tracking for a WMR”
2. Isidori A, Byrnes CI. Output regulation of nonlinear systems. IEEE Trans Autom Control, 1990; 35(3): 131–140. 3. Hermann R, Krener AJ. Nonlinear controllability and observability. IEEE Trans Autom Control, 1977; 22(5): 728–740. 4. Astolfi A, Karagiannis D, Ortega R. Nonlinear and Adaptive Control with Applications, Springer Verlag, 2008.
Final Comment by the Authors M. Velasco-Villa, E. Aranda-Bricaire, H. Rodríguez-Cortés, J. González-Sierra The authors warmly thank M. Ishikawa, D. Carnevale and F. Martinelli for the thorough reading, extensive simulations, and detailed comments that they have gifted to our paper.
Comment to the discussion by M. Ishikawa A realistic situation where the absolute position and velocity of the robot are measurable while the attitude angle is not would be, for instance, the case where a satellitebased GPS or an indoor ultrasonic positioning system is employed. In the case that the set T is finite, the fact that u1 = 0 implies that V converges to zero asymptotically, but perhaps not exponentially. The Authors agree with Ishikawa’s point in the sense that the sampling period may have a dramatic impact in the closed-loop system behavior under noisy measurements.
Comment to the discussion by D. Carnevale, F. Martinelli The suggestion put forward by Carnevale and Martinelli to cope with the case of compact time intervals T is most welcome and will be addressed in future work. The use of the Euler approximation and the EKF to estimate the attitude angle as well as the longitudinal and angular velocities deserves a more detailed response. First, of course, there is no attempt on our part to diminish the
merits of the celebrated EKF. Our only point in that our observer-based approach allows to provide formal convergence Proofs, while this is not necessarily the case when the EKF is employed. Second, since the ultimate goal of Carnevale and Martinelli’s extensive simulations is to provide a fair comparison between the observer- and EKF-based strategies, perhaps the use of an exact discrete-time model, instead of Eulers’ would be more suitable (see for instance [1, 2, 3]). Third, while the study of the performance of both schemes subject to noisy measurements seems natural, the Authors claim that the pollution of control signals by noise is less reasonable in the present context. For one reason, because the “integration” of the acceleration input v1 to obtain u1 takes place inside of the control device only, but not as a physical process. For another, because the actuators of virtually all mobile robots perform inner control loops which render the velocity errors negligible.
References 1. Aranda Bricaire E, Moog CH. Linearization of discrete-time systems by exogenous dynamic feedback. Automatica, 2008; 44(7): 1707–1717. 2. Velasco-Villa M, Aranda-Bricaire E, Orosco-Guerrero R. Discrete-Time Modeling and Path-Tracking for a Wheeled Mobile Robot, Computación y Sistemas Rev Iberoam Comput, 2009; 13(2): 142–160. 3. Niño-Suárez PA, Aranda-Bricaire E, Velasco-Villa M. Discrete-Time Sliding Mode Path-Tracking Control for a Wheeled Mobile Robot. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, USA, December 13–15, 2006, pp. 3052–3057.
Discussions on: “Observer Based Trajectory Tracking for a WMR”
Discussion on: “Trajectory Tracking for a Wheeled Mobile Robot Using a Vision Based Positioning System and an Attitude Observer” Masato Ishikawa Dept. of Mechanical Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan
1. Quick Summary This paper by Velasco-Villa et al. [1] proposes an observerbased tracking control method for a wheeled planer mobile robot. The proposed method is simply summarized as follows.
from which the estimate for θ is uniquely reconstructed. Output feedback control is realized by replacing the unmeasurable information in the controller with the estimates.
2. Discussion
State Equation x˙ 1 = u1 cos θ,
θ˙ = u2 ,
x˙ 2 = u2 sin θ,
u˙ 1 = v1
where (x1 , x2 ) is the robot’s position, θ is the attitude angle and u1 is its velocity. The control input is U = (v1 , u2 )T where v1 is the acceleration and u2 is the angular velocity. A key assumption is that θ is not measurable. Tracking controller The state vector is split into y1 = (x1 , x2 )T and y2 = (θ, u1 )T . Let y1d and y2d be the respective reference trajectories, where y2d is dependent on y1d so that they satisfy the state equation, and y1e , y2e be the respective tracking error. The controller consists of a pair of P/PD feedback laws based on feedback linearization: −1 cos θ −u1 sin θ (¨y1d − Kd1 y˙ 1e − Kp1 y1e ) U1 = sin θ u1 cos θ −1 0 1 (˙y2d − Kp2 y˙ 2e ) U2 = 1 0 Choose U = U1 if |u1 (t)| ≥ δ, or U = U2 otherwise. Attitude observer
2.1. On Measurement Setting
η˙ 1 = −(η2 + x2 arctan u1 ) −u1 arctan u1 [η1 + arctan u1 ] −
x1 v1 1 + u12
η˙ 2 = −(η1 + x1 arctan u1 ) −u1 arctan u1 [η2 + arctan u1 ] −
x2 v1 1 + u12
where is the observer gain. η1 estimates (cos θ − x1 arctan u1 ) while η2 estimates (sin θ − x2 arctan u1 ), E-mail: [email protected]
Observer design and output feedback control for mobile robots is a challenging topic from both theoretical and practical points of view. A major difficulty is that the system is inherently nonlinear, thus the observer design and controller design is not simply separated as in the case of linear systems [2]. In particular, observability of the system is often spoiled when the robots remains stationary, or u1 = 0. This exemplifies a fundamental property of nonlinear systems that observability relies on control inputs [3], which makes things seriously awkward. With this respect, this paper proposes to combine 1) a switched controller to get out from the stationary (i.e., unobservable) states, with 2) an observer which is capable to estimate the attitude angle in spite of the spontaneous u1 . I read this paper with great interest. This paper provides a simple and practical example of observer design scheme based on the immersion and invariance (I&I) approach proposed by Astolfi et al. [4], followed by some physical experiments to prove its effectiveness. The I&I methodology allows us to consider indirect estimates η1 , η2 to obtain θ , which seems to be a promising approach for robust estimation. For all these contributions, I raise several points to be discussed.
First, I wonder which sort of realistic situation requires the problem setting, that the robot’s absolute position and velocity is measurable while the attitude angle is unmeasurable. Any example that supports this problem setting would be helpful. 2.2. On Convergence of the Estimation Error The authors assume that T := {t ∈ R|u1 (t) = 0} is a set of isolated time instants, namely, it does not contain nonempty interval nor accumulation point. Under
361
Discussions on: “Observer Based Trajectory Tracking for a WMR”
Fig. 2. Simulation under measurement noise with faster sampling.
• The tracking error y2e converges to zero as long as |u1 (t)| < δ holds forever. This is true. However, we should note that the two modes are switched to each other when |u1 (t)| cross the threshold δ, so the state may not remain in a single mode. Influence of switching should be taken into account to guarantee the whole convergence. This leads us to take closer look at the following cases. Fig. 1. Simulation under measurement noise with slower sampling.
this assumption, Proposition 1 claims convergence of the Lyapunov candidate V1 (t) to zero. Here the point we should take care is that V˙ 1 (t) is timedependent and vanishes when u1 (t) = 0. • Even if T is a finite set, V1 (t) does not necessarily converge to zero, because u1 (t) → 0 (t → ∞) may occur. I suggest this case should be excluded in advance by assumption. • For the case where T is infinite, the authors claim that S := V1 (T ) converges to zero because it is nonnegative and monotonically decreasing. In general, however, a nonnegative and monotonically decreasing series may converge to a nonzero limit. I suppose the conclusion itself can be justified by appropriately modifying the proof.
2.4. On Robustness and Possible Influence of Switching Fig. 1 shows a simulation result of the proposed scheme with ±5[mm] of measurement noise is imposed on x1 and x2 . The sampling period is set to 10[msec]. Other design parameters and the reference trajectory y1d = (0.0 sin t, 1 − 0.5 cos2 t)T are almost similar to the ones used in the paper. It seems to work quite nice in spite of the presence of measurement noise. However, if we shorten the sampling period to 1[msec], it starts to chatter around the threshold |u1 (t)| = δ as shown in Fig. 2. The chattering often occurs when the input v1 (t) takes difference signs in the both sides of the threshold, mainly because of the presence of measurement noise. We might overlook this phenomena when the sampling is relatively slow, but it is noteworthy that faster sampling (or faster switching) tend to make the system sensitive to noise.
2.3. On Convergence of the Tracking Error As for the stability of observer-controller combination, in Proposition 2, the authors proved that • The tracking error y1e converges to zero as long as |u1 (t)| ≥ δ holds forever.
Acknowledgment The author of this discussion is grateful to Prof. Teruyo Wada for her variable comments on Lyapunov stability and convergence analysis.
362
References 1. Velasco-Villa M, Aranda-Bricaire E, Rodrígues-Cotés H, González-Sierra J. Trajectory tracking for a wheeled mobile robot using a vision based positioning system and an attitude observer. Eur J Control, 2012; to appear.
Discussions on: “Observer Based Trajectory Tracking for a WMR”
2. Isidori A, Byrnes CI. Output regulation of nonlinear systems. IEEE Trans Autom Control, 1990; 35(3): 131–140. 3. Hermann R, Krener AJ. Nonlinear controllability and observability. IEEE Trans Autom Control, 1977; 22(5): 728–740. 4. Astolfi A, Karagiannis D, Ortega R. Nonlinear and Adaptive Control with Applications, Springer Verlag, 2008.
Final Comment by the Authors M. Velasco-Villa, E. Aranda-Bricaire, H. Rodríguez-Cortés, J. González-Sierra The authors warmly thank M. Ishikawa, D. Carnevale and F. Martinelli for the thorough reading, extensive simulations, and detailed comments that they have gifted to our paper.
Comment to the discussion by M. Ishikawa A realistic situation where the absolute position and velocity of the robot are measurable while the attitude angle is not would be, for instance, the case where a satellitebased GPS or an indoor ultrasonic positioning system is employed. In the case that the set T is finite, the fact that u1 = 0 implies that V converges to zero asymptotically, but perhaps not exponentially. The Authors agree with Ishikawa’s point in the sense that the sampling period may have a dramatic impact in the closed-loop system behavior under noisy measurements.
Comment to the discussion by D. Carnevale, F. Martinelli The suggestion put forward by Carnevale and Martinelli to cope with the case of compact time intervals T is most welcome and will be addressed in future work. The use of the Euler approximation and the EKF to estimate the attitude angle as well as the longitudinal and angular velocities deserves a more detailed response. First, of course, there is no attempt on our part to diminish the
merits of the celebrated EKF. Our only point in that our observer-based approach allows to provide formal convergence Proofs, while this is not necessarily the case when the EKF is employed. Second, since the ultimate goal of Carnevale and Martinelli’s extensive simulations is to provide a fair comparison between the observer- and EKF-based strategies, perhaps the use of an exact discrete-time model, instead of Eulers’ would be more suitable (see for instance [1, 2, 3]). Third, while the study of the performance of both schemes subject to noisy measurements seems natural, the Authors claim that the pollution of control signals by noise is less reasonable in the present context. For one reason, because the “integration” of the acceleration input v1 to obtain u1 takes place inside of the control device only, but not as a physical process. For another, because the actuators of virtually all mobile robots perform inner control loops which render the velocity errors negligible.
References 1. Aranda Bricaire E, Moog CH. Linearization of discrete-time systems by exogenous dynamic feedback. Automatica, 2008; 44(7): 1707–1717. 2. Velasco-Villa M, Aranda-Bricaire E, Orosco-Guerrero R. Discrete-Time Modeling and Path-Tracking for a Wheeled Mobile Robot, Computación y Sistemas Rev Iberoam Comput, 2009; 13(2): 142–160. 3. Niño-Suárez PA, Aranda-Bricaire E, Velasco-Villa M. Discrete-Time Sliding Mode Path-Tracking Control for a Wheeled Mobile Robot. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, USA, December 13–15, 2006, pp. 3052–3057.