145 Pages·2001·5.67 MB·English

1 Model Development and Control Objectives 1.1 Introduction In this chapter, we describe the kinematic model for wheeled mobile robots (WMRs) for the so-called kinematic wheel under the nonholonomic con- straint of pure rolling and non-slipping. Based on the kinematic model, we present differentiable, time-varying kinematic controllers for the regulation and the tracking control problems. Through a Lyapunov-based stability analysis, we demonstrate that the controllers yield global asymptotic reg- ulation or tracking and that all signals remain bounded during closed-loop operation. From the stability analysis, it is clear that the regulation problem cannot be solved as a special case of the tracking problem due to restric- tions on the desired trajectory. Motivated by the facts that the kinematic controllers are limited to asymptotic results and that the tracking controller does not solve the regulation problem as a special case, we introduce a uni- fled kinematic control structure (i.e., the regulation problem is a special case of the tracking control problem). Through a Lyapunov-based stability analysis, we demonstrate that the unified kinematic controller yields global exponential tracking and regulation and that all signals remain bounded during closed-loop operation. In addition to the kinematic control problem, we also examine incorporat- ing the dynamic model in the overall control design. To this end, we present the dynamic model and describe several associated properties. Based on the 2 .1 Model Development and Control Objectives dynamic model, we investigate the use of standard backstepping techniques to develop a torque control input for the unified control problem (versus the velocity control input for kinematic controllers). Through a Lyapunov- based stability analysis, we demonstrate that the unified controller yields global exponential regulation; however, we illustrate how the proposed de- sign breaks down for the tracking problem. In subsequent chapters, we illustrate how the unified kinematic controller can be redesigned to incor- porate the effects of the dynamics via standard backstepping techniques. 1.2 Kinematic Model Development The kinematic model for the so-called kinematic wheel under the nonholo- nomic constraint of pure rolling and non-slipping is given as follows ]71[ = s(q)v (1.1) where q(t), O(t) 3 E R are defined as )2.1( xc(t) and yc(t) denote the position of the center of mass (COM) of the WMR along the X and Y Cartesian coordinate frames and O(t) E 1 R represents the orientation of the WMR (see Figure 1.1), ~s(t) and ~¢(t) denote the Cartesian components os0 of [ the linear ]0 velocity, denoted by ~v (t) E R] ,1 0(t) E 1~] denotes the angular velocity, the matrix S(q) E 2×3~] is defined as follows S(q)= sin(cid:127) 0 (1.3) 0 1 and the velocity vector v(t) 2 E R is defined as V = V[ 1 v2]T = [vl ]O T • (1.4) Note that the COM and the center of rotation are assumed to coincide. 1.3 Regulation Problem In this section we present the open-loop error system for the regulation problem. Based on the open-loop error system, we present the differen- tiable, time-varying feedback controller given in [20] and then examine the stability of the resulting closed-loop error system through a Lyapunov- based stability analysis. 3.1 Regulation Problem 3 cY c X X ~ Figure 1.1. Wheeled Mobile Robot 1.3.1 Open-Loop Error System The control objective for the regulation problem is to force the actual Cartesian position and orientation to a constant reference position and orientation. To quantify the regulation control objective, we define k(t), ~(t), 0(t) E ~i as the difference between the actual Cartesian position and orientation and the reference position and orientation as follows ~ = x~ - x~ ~ = y~ - Yr~ 0 = 0 - 0~ (1.5) where x~(t), yc(t), O(t) were defined in (1.2) and x~c, yr~, ~0 1 E N represent the constant reference position and orientation. To facilitate the closed- loop error system development and stability analysis, a global invertible transformation was defined 0sion c[ [20] as foll ows 0nis (1.6) 2e = -sinO cosO 0 3e 0 0 1 to relate the auxiliary error signals denoted by el(t), e2(t), e3(t) E 1 R to the position and orientation regulation error signals ~(t), ~(t), 0(t) defined 4 .1 Model Development and Control Objectives in (1.5). After taking the time derivative of (1.6) and using (1.1-1.4), the kinematic model given in (1.1) can be rewritten in terms of the auxiliary variables defined in (1.6) as follows (1.7) e2 "-~ --v2el • a~ v2 Remark 1.1 Based on the inverse of the transformation defined in (1.6) given as follows 0soc[ 0nis 0][ex] j~ = sin 0 cos ~ 0 e2 (1.8) 0 0 1 ea it is clear that if t~m el (t), e2 (t), e3(t) = ,O then miloo.--t ~(t), ~l(t), ~(t) = .O 1.3.2 Control Development The regulation control objective is to design a controller for the transformed kinematic model given by (1.7) that forces the actual Cartesian position and orientation to a constant reference position and orientation. Based on this control objective, a differentiable, time-varying controller was proposed in [20] as follows v2 -k2e3 + e~ sin(t) (1.9) where kl, k2 E 1~] are positive constant control gains. After substituting (1.9) into (1.7) for vl(t) and v2(t), the following closed-loop error system is obtained ~ = -v2el • (1.10) e3 -k2e3 + 2 e sin(t) Remark 1.2 Note that the closed-loop dynamics for e3(t) given in (1.10), represent a stable linear system subjected to an additive disturbance given by the product e2(t)sin(t). If the additive disturbance is bounded 5.e., if e2(t) C Coo), then it is clear that e3(t) C/:oo- Furthermore, if the additive disturbance asymptotically vanishes 5.e., if oomLFt e2(t) = O) then it is clear from standard linear control arguments [7] that Um e3 (t) = .O t ~--- O0 3.1 Regulation Problem 5 1.3.3 Stability Analysis Given the closed-loop error system in (1.10), we can now invoke Lemma A.2, Lemma A.12, and Lemma A.14 of Appendix A to determine the sta- bility result for the kinematic controller given in (1.9) through the following theorem. Theorem 1.1 The differentiable, time-varying kinematic control law given in (1.9) ensures global asymptotic position and orientation regulation in the sense that (i.II) lira 5:(t), (cid:127)(t), O(t) = .O *-..'-t CO Proof: To prove Theorem i.I, we define a non-negative function denoted by 1/1 (t) e ~1 as follows + After taking the time derivative of (1.12), substituting (1.10) into the re- suiting expression for el(t) and e2(t), and then cancelling common terms the following expression is obtained =-kle . )31.1( Based on (1.12) and (1.13), it is clear that el(t), e2(t) E /:oo. Since e2(t) E/~oo it is clear from Remark 1.2 that e3(t) E/:oo. Based on the fact that el(t), e2(t), e3(t) E/:oo, we can utilize (1.9) and (1.10) to prove that vl(t),v2(t),el(t),e2(t),e3(t) e ~oo. Since ~l(t), e2(t), e3(t) e /:o~, we can invoke Lemma A.2 of Appendix A to conclude that el(t), e2(t), e3(t) are uniformly continuous. After taking the time derivative of (1.9) and utilizing the aforementioned facts, we can show that ~)l(t), /J2(t) C/:oo, and hence, vl(t) and v2(t)are uniformly continuous. We have already proven that el(t), ~l(t) E £0o. If we can now prove that el(t) E/:2, we can invoke Leinma A.12 of Appendix A to prove that lim el(t) ---- .O (1.14) t +---- ~C To this end, we integrate both sides of (1.13) as follows 5/ 0/ Vl(t)dt = kl e~(t)dt. (1.15) - After evaluating the left-side of (1.15), we can conclude that F kl e2(t)dt = 1V (0) - 1V (co) < 1/1 (0) < c~ (1.16) 6 .1 Model Development and Control Objectives where we utilized the fact that VI(0) _> 1V (oo) > 0 (see (1.12) and (1.13)). Since the inequality given in (1.16) can be rewritten as follows 0/i 4(t)dt < V < o~ (1.17) we can utilize Definition A.1 to conclude that et(t) E £2. Based on the facts that el(t),~l(t) e L:oo and el(t) E /:2, we can now invoke Lemma A.12 of Appendix A to prove the result given in (1.14). After taking the time derivative of the product el(t)e2(t) and then sub- stituting (1.10) into the resulting expression for the time derivative of el (t), the following expression is obtained d t-d (ele2) = [e~v2] + el (e2 -- kle2). (1.18) Given the facts that lim el(t) = 0 and the bracketed term in (1.18) is t--*OO uniformly continuous (i.e., e2(t) and v2(t) are uniformly continuous), we can invoke Lemma A.14 of Appendix A to conclude that lira d (el(t)e2(t)) = 0 lira e~(t)v2(t) = 0. (1.19) t ---*oO a~: t ---+oo From (1.19), it is clear that lira e2(t)v2(t) = 0. (1.20) After utilizing (1.9), (1.10), (1.14), and (1.20), we can conclude that lim vl(t) = 0 lim ~l(t) = 0 lira ~2(t) = 0. (1.21) t---*OO t---*oo t~Oo To facilitate further analysis, we take the time derivative of the product e2(t)v2(t) and utilize (1.7) and (1.9) to obtain the following expression ~/(~v~) d = [4 cos(t)] + ~ (v~ + 24 sin(t)) - k~v~. (1.22) Since the bracketed term in (1.22) is uniformly continuous, we can utilize (1.20) and (1.21) and invoke Lemma A.14 of Appendix A to conclude that t--.oo lim d dt (e2(t)v2(t)) = 0 mil~o.-t e~(t) cos(t) = 0. (1.23) From the second limit in (1.23), it is clear that lira e2(t) = 0. (1.24) t OO~---- Based on (1.24), it is clear from Remark 1.2 that lim e3(t) = 0. (1.25) t--*OO After utilizing (1.8), (1.14), (1.24), and (1.25), we obtain the global asymp- totic regulation result given in (1.11). • 4.1 Tracking Problem 7 1.4 Tracking Problem For the previous regulation problem, the control objective was to force the actual Cartesian position and orientation to a constant reference position and orientation. In contrast to the regulation problem, the control objective for the tracking control problem is to force the actual Cartesian position and orientation to track a time-varying reference trajectory. To quantify the tracking control objective, we define ~(t), ~(t), 0(t) I C R as follows = xc - xrc = Y cY - crY 0 = 0 - rO (1.26) where the actual position and orientation, denoted by xc(t), yc(t), 0(t), were defined in (1.2) and qr (t) = [ x~c(t) y~(t) Or(t) ]T C (I1 3 denotes the time-varying reference position and orientation. In order to ensure that the reference trajectory is selected to satisfy the pure rolling and nonslip- ping constraint imposed on the actual WMR, the reference trajectory is generated via a reference robot which moves according to the following dynamic trajectory rO = S(q~)vr (1.27) where S(-) was defined in (1.3) and v~(t) = [ vlr(t) v2r(t) ]T E 2 R denotes the reference time-varying linear and angular velocity. With regard to (1.27), it is assumed that the signal v~(t) is constructed to produce the desired motion and that v~(t), iJr(t), qr(t), and 0r(t) are bounded for all time. Remark 1.3 To illustrate one method for selecting vr(t) such that a de- sired Cartesian path is generated, we first express the desired trajectory as follows yrc(t) = gr(Xrc(t)) )S2.1( where gr (') E 1 R is a desired path selected to eb second order differentiable. To facilitate further analysis, we divide the first row of (1.27) by the sec- ond row and perform some algebraic manipulation to obtain the follovnng expression rgO -- tan rO (1.29) crXO where (1.28) was utilized. After taking the time derivative of (1.29), we obtain the follougng expression d ( rgO ~ 02gr c~& = (1 + tan 2 0r) 0r- (1.30) 8 .1 Model Development and Control Objectives After substituting (1.27) into (1.30) for &~c(t) and then rearranging the resulting expression, we obtain the following expression I {02gr~ a (1.31) 0r - (1 +tane0r) \~x2rc] vlrcosOr =V2r where vlr(t) represents the desired linear velocity which can eb arbitrarily selected. Based on (1.28) and (1.29), it is clear that at(O) should eb selected as follows )23.1( yrc(0) = gr(x~o(0)) 0r(0) 0--~rclxrc(0) = ta-nI where xrc( ) O is arbitrarily selected. 1.4.1 Open-Loop Error System To develop the open-loop tracking error system, we take the time deriv- ative of (1.6) and use (1.1-1.4), (1.26), and (1.27) to obtain the following expression [20] 2ev+lv[ oeriv ]3e e2 = --v2el -4- Vlr Sin e3 (1.33) 3}~ v2 -- v2r. Remark 1.4 Note that if Vlr(t), v2r(t) = 0, then the open-loop regulation error system given in (1.7) is recovered. 1.4.2 Control Development The tracking control objective is to design a controller for the transformed kinematic model given by (1.33) that forces the actual Cartesian posi- tion and orientation to track the timewarying reference trajectory given in (1.27). Based on this control objective, a differentiable time-varying con- troller was proposed in [20] e3, [ as folklel+vlcos lows ] = sin e3 (1.34) 2 V --Vlr e3 2 e -- k2e3 -F v2r where kl, k2 E 1 E are positive constant control gains. After substituting (1.34) into (1.33) for vl(t) and v2(t), we obtain the following closed-loop 1.4 Tracking Problem 9 error system v2e2 - klel -v2el q- wit sine3 (1.35) sin 3 e --Vlr e 2 -- k2e3. e3 1.4.3 Stability Analysis Given the closed-loop error system in (1.35), we can now invoke Lemma A.2, Lemma A.12, and Lemma A.13 of Appendix A to determine the stability result for the kinematic controller given in (1.34) through the following theorem. Theorem 1.2 Provided the reference trajectory (i.e., v~( t ), iJr( , ) t q~( , ) t and O~(t)) is selected to be bounded for all time and that lim vlr(t) ~ 0, (1.36) t---*Oo the kinematic control law given in (1.3~) ensures global asymptotic position and orientation tracking in the sense that lim ~(t), ~(t), ~(t)= 0. (1.37) $~OO Proof: To prove Theorem 1.2, we define a non-negative function denoted by V2(t) E ~1 as follows 12 12 2V = ~e½ + ~e2 + ~e 3. (1.38) After taking the time derivative of (1.38), substituting (1.35) into the re- sulting expression for ~l(t), ~2(t), and ~3(t), and then cancelling common terms, we obtain the following expression V2 = -kle~ - k2e~. (1.39) Based on (1.38) and (1.39), it is straightforward that el(t), e2(t), e3(t) E /:co and that el(t),e3(t) E £2 (see (1.15-1.17)). Since el(t), e2(t), e3(t) e £~,we can utilize (1.34), the assumption that v~(t) E /:~, and the fact that sin e3 lim ~ = 1 (1.40) e3---*0 e 3 to prove that vl(t), v2(t) E/~. From these facts, we can utilize the closed- loop tracking error dynamics given in (1.35) to prove that ~l(t), ~2(t), ~3(t) ~ L~; hence, by invoking Lemma A.2 of Appendix A, it is clear 10 .1 Model Development and Control Objectives that el(t), e2(t),e3(t) are uniformly continuous. Since el(t), ~l(t), e3(t), ~3(t) E/:oo and el(t), e3(t) e £:2, we can invoke Lemma A.12 of Appendix A to conclude that lira el(t) ---- 0 lira e3(t) ---- .O (1.41) t--~O0 QO*---$ In order to prove that lim e2(t) = 0, we first take the time derivative of OO*--t the closed-loop tracking error dynamics for e3(t) given in (1.35) as follows e3 -- -~)lr 3e3enis e2 - vlr 3e3enis "2e - vlr ,k//e3 - coSeae~ sine3) e3e2 - k2~3. (1.42) Based on the fact that e2(t), ~2(t), e3(t), ~3(t) e/:oo, the assumption that v~(t), 9~(t) e/:oo, and the fact that lira ¢/ e3 cos e3 --- sin e3 ) (1.43) 3-0 \ e] = 0 we can conclude from (1.42) that ~3(t) C £oo; hence, from Lemma A.2 of Appendix A, ~3(t) is uniformly continuous. Based on (1.41) and the fact that ~3(t) is uniformly continuous, we can use the following equality lim /0 ~ d (e3(T)) dT= lim e3(t) + Constant (1.44) OO*---t ~TT oo*----t to conclude that the left-side of (1.44) exists and is finite; hence, we can now invoke Lemma A.13 of Appendix A to prove that lira 3(t) = o. (1.45) Oo*---t Based on (1.41) and (1.45), it is straightforward from (1.35) that t~oo lim vlrt ,t )~s~inee32(t[) "t" ) = 0. (1.46) Finally, based on (1.36) and (1.40), we can conclude from (1.46) that nm e2(t) = o. (1.47) Based on (1.8), (1.41) and (1.47), the global asymptotic tracking result given in (1.37) can now be directly obtained. • Remark 1.5 Based on the restriction placed on the reference trajectory given in (1.36), the regulation problem described in Section 1.3 cannot eb solved with the tracking controller given in (1.34).

This book examines the control problem for wheeled mobile robots. Several novel control strategies are developed and the stability of each controller is examined utilizing Lyapunov techniques. The performance of each controller is either illustrated through simulation results or experimental results

Upgrade Premium

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.