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1 998 Technical Report Driftless [9] driftless ( ) driftless, P θ η P dη/ R l, R r W ω l, ω r P θ P dη/ d/, d/ d = dη d cos θ, = dη sin θ () d = d tan θ () u, u dη/ dθ/ u = dη u = dθ = R lω l + R r ω r = R lω l + R r ω r W () (3) (3-a) (3-b) d = cos θ sin θ u + u (4) θ ξ =(,, θ) T = cos ξ 3 sin ξ 3 u + u = f (ξ)u + f (ξ)u (5) (drift A ) driftless 3 ξ (, ) =(, ) θ =

2 P θ u (t) = u (t) = ( t<t ) u (t) = u (t) = (t t<t + t ) u (t) = u (t) = (t + t t<t + t ) u (t) = u (t) = (t + t t<t +t ) (8) η Figure : 3 (5) = f ()u + f ()u + O (ξ,u,u ) = u + u + O (ξ,u,u ) (6) d = f ( )u + + f m ( )u m (7) n m driftless sstem 3 (5) (,, θ) =(,, ) (5) n ξ = ξ u =, u = ξ = f (ξ ) f (ξ ) u =,u = f (ξ ) ξ = ξ f (ξ ) f (ξ ) ξ() = ξ t t = f (ξ) ( t<t ) = f (ξ) (t t<t + t ) = f (ξ) (t + t t<t + t + t ) = f (ξ) (t + t + t t<t + t + t + t ) d ξ = d = f ξ (9) = f() () = d f(ξ) = f f(ξ) () ξ t ξ(t) =ξ() + t + d ξ ξ() t + O 3 (t) ξ() = ξ() + f(ξ())t + f ξ f(ξ())t ξ() +O 3 (t) () n f(ξ) i f (i) (ξ) f ξ f () f () f ξ ξ () ξ n f f ξ = () f () f ξ ξ () ξ n (3) f (n) ξ f (n) ξ d ξ = d = f ξ f (n) ξ n = f() (4) = d f(ξ) = f f(ξ) (5) ξ t ξ(t) =ξ() f(ξ())t + f ξ f(ξ())t ξ() + O 3 (t) (6)

3 (9) ξ() = ξ t t ξ(t ) ξ(t )=ξ + f (ξ )t + f ξ ξ := ξ + f (ξ )t + ξ(t + t ) f (ξ )t ξ f ξ + O 3 (t ) (7) f (ξ )t (8) ξ ξ(t + t )=ξ(t )+f (ξ(t ))t + f ξ f(ξ(t ))t + O3 (t,t ) ξ(t) = ξ + f (ξ )t + f ξ f (ξ )t ξ +f (ξ + f (ξ )t + f ξ f (ξ )t )t ξ + f ξ f (ξ )t + O3 (t ) ξ = ξ + f (ξ )t + f ξ f (ξ )t + f (ξ )t ξ + f ξ f (ξ )t t + f ξ ξ f(ξ )t ξ +O 3 (t ) (9) ξ(t + t + t + t )= ( f ξ + ξ f (ξ ) f ) ξ ξ f (ξ ) t t + O 3 (t,t ) ξ () ξ() = ξ f ξ f (ξ ) f ξ ξ f (ξ ) () ξ Lie bracket Lie bracket [f,f ](ξ) [f,f ](ξ) = f ξ f (ξ) f ξ f (ξ) () [f,f ](ξ ) ξ() = ξ f (ξ ), f (ξ ), [f,f ](ξ ), [f, [f,f ](ξ ), n f (ξ) = cos(ξ 3) sin(ξ 3 ) f (ξ) = [f,f ](ξ) = f ξ f (ξ) f ξ f (ξ) = f (ξ) sin(ξ 3) cos(ξ 3 ) = sin(ξ 3) cos(ξ 3 ) 3 33 [8] (5) u = γ (ξ), u = γ (ξ) (3) γ i (ξ) ξ γ i (ξ) ξ (5) (3) = f (ξ)γ (ξ)+f (ξ)γ (ξ) (4) ξ f, f δ R n = f (ξ)γ (ξ)+f (ξ)γ (ξ)+δ (5) ξ δ ξ δ =f (ξ δ )γ (ξ δ )+f (ξ δ )γ (ξ δ )+δ (6) 3

4 f (ξ δ )u + f (ξ δ )u = δ (7) ξ δ u, u ξ δ u, u (Brockett[3]) d/ = f(, u) R n, u R m f(, ) = f(, u) =, u = = N R n u = N u R m N R n δ N f(, u) =δ, u N, N u (5) δ =(δ,δ,δ 3 ) T cos ξ 3 sin ξ 3 u + u = δ δ δ 3 (8) ξ u, u ε δ = 3 ε, δ = ε (9) u ξ 3 u = ε, ξ 3 = π 3 (3) ξ 3 (5) (7) n m { f () f m ()} driftless sstem 4 4 Chained form driftless chained form [8] ż = g (z)v + g v (3) g (z) = z, g = z n chained form Chained form µ = v, µ = v v (3) z n z n d z n z n dz = + µ (33-a) z 3 z z dz = µ (33-b) t z z [][] chained form [] chained form [][] 4

5 (5) chained form z = ξ z =tanξ 3 z 3 = ξ u = v cos ξ 3 u =cos ξ 3 v z = v z = v z 3 = z v (34-a) (34-b) (34-c) (35-a) (35-b) (36-a) (36-b) (36-c) µ = v (37-a) µ = v v (37-b) ( ) ( ) ( ) d z3 z = + µ dz z (38-a) dz = µ (38-b) π ξ 3 π 4 [][] (38-a) µ = k z k 3 z 3 (39) (38-b) µ z (38-a) µ z t z, z 3 z z = z z d dz ( z3 ) = z ( z ) ( ) µ (4) µ = k z k 3 z 3 (4) (4) z z z, z 3 z v { k z v = v k 3 z 3 v,v > (ż > ) k z v k 3 z 3 v, v < (ż < ) = k 3 z 3 v k z v (4) z z λ> v = λz z = z z, z 3, θ, θ ( ) µ (, θ ) 43 Chained form 43 Sordalen K Sordalen and Egeland [5] chained form K- K- λ class K ζ( ) ζ() = z(t) ζ( z() )e λt H> z(t) H z() e λt H z() ζ( z() ), <K sat(, K) = K, > K K, < K {, ( ) sgn() :=, ( <) 5

6 v T> t i {,, } t i := it k( ) :R n R: z k(z) ; K > z R n k(z) K, z = k(z) = 3 T f( ) :R + R: t f(t) P) [t, + ) P) f(t), t t P3) i {,, } f(t i )= P4) j {3,,n} η j >,P j >, p {,, }, t t p t [f j 3 (τ) η j ]dτ t p P j 4 v = k(z(t i ))f(t), t [t i,t i+ ) (43) Sordalen k(z), f(t) ( cos ωt) f(t) =, ω = π T (44) k(z) =sat( [z +sgn(z )G( z )]β,k), (45) G( z )=κ z n 4 β = ti+ t i f(τ)dτ κ -norm n z := z j v j= λ,,λ n {g jm ; j, m =,,n} g n,n = λ n g j,m (t) =g jm {λ j f j (t)+(j ) f(t)} +f(t){ġ jm (t)+g j,m+ (t)f(t)} g j,j (t) = λ j + f (t)g j,j+ (t) g jp = ifp j or p = n + 3 v = { Γ(k(z(t )),t) T Z, z(t i ), z(t i )= (46) n Γ(k,t) = [Γ (k,t),, Γ n (k,t)] Γ (k,t)= λ + f 3 g,3 Γ j (k,t)=f(λ fg j + fg j + fġ j + f g,j+ ) k j (f g jm t ) v () Z v v f(t) Z v =Γ(t)Z z v k(z(t i )) v Z Γ(t) 3 Z λ,,λ n v K f(t) k(z(t i )) z κ v Periodic generator Gain function f(t) = cos t (47) k(z) =sat( [z +sgnz G( z )]β,k) (48) K v g- λ = λ = λ = λ 3 (49) g,3 = λ (5) Γ= [ Γ Γ 3 ] (5) 6

7 3 Γ = λ + f(t) 3 g,3 (5) = λ + f(t) 3 ( λ) = λ( + f(t) 3 ) Γ 3 = f(t) k(z) ( λ f(t) λ f(t)) (53) v =Γ(k(z(t i )),t)z (54) 43 Samson Samson[4] chained form Skewsmmetric chained form z(t) H z() e(t) H e(t) Skew-smmetric chained form v = k(z(it ))h(t), t [it, (i +)T ) (55-a) v = (λ + λ h(t) 3 )z (55-b) + h(t) k(z(it )) ( λ λ h(t) λ ḣ(t))z 3 χ = z χ = z n χ 3 = z n χ 4 = k z + L g z 3 χ j+3 = k j z j+ + L g z j+ (58) Γ = λ λ 3 h(t) 3 (56) Γ 3 = h(t) k(z) ( λ λ 3 h(t) λ 3 ḣ(t)) (57) h(t) = cos(πt/t), k(z) =sat( [z +sgnz G(z)]/T, K) j =,,n 3 χ χ = v χ = v χ 3 = k χ v + χ 4 v χ j+3 = k j+ χ j+ v + χ j+4 v, j =,,n 4 χ n = k n χ n v (59) G(z) =κ( z + z + z 3 ) T,K,κ,λ,λ v z k(z) G(z) v h(t) T z k(z) G(z) z G(z) v z v z z z v z, z 3 λ,λ z,z 3 v h(t), k(z(it )) z z 3 z K v = (k n χ n + L g χ n )v + w χ n χ n = k n χ n v + w (6) v w 3 v v = k v χ + h(χ,,χ n,t) (6) k v > h( ) h(,,,t)= 4 w k w > w = k w v χ n (6) 7

8 ( ) f(t) K z(t) K z() f(t) ξ = χ h Z Skew-smmetric chained form χ = z χ = z 3 χ 3 = z (63) n =3z z 3 3 v = k χ v + w χ 3 = k z v + w v = k v z + h(z,t) w = k w v χ 3 v = k v z +(z + z 3 )sin(πt/t) v = k z 3 v k w z v (64-a) (64-b) (65-a) (65-b) T>, k >,k v >,k w > v k v z z (z + z 3)sin(πt/T) z, z 3 z v (4) z z, z 3 z k Skew-smmetric chained form 433 Pomet Pomet[] Pomet[] ( drift-free sstem chained sstem ) Time-varing Controller sstematic T h(t, z,z 3,,z n ) h(t, ) = V (t, z) = z + (z + h(t, z,z 3,,z n )) + z z n (66) α(t, z) = h t (t, z,z 3,,z n ) (67) v = L g V v = α(t, z) L g V (68-a) (68-b) L gi V V g i Lie L gi V = V z g i (66) h(t, z,z 3,,z n )=z cos t α(t, z) = z sin t V (t, z) = z + (z + z 3 cos t) + z 3 (69) V v = (z + z 3 cos t)z cos t (z z 3 + z ) v = z 3 sin t (z + z 3 cos t) (7-a) (7-b) (69) z Sordalen Samson 44 8

9 44 Khennouf and Canudas de Wit [5][6][6] z = z = V (z) =z + z s(z) =z 3 z z (7-a) (7-b) σ >κ κ, σ v = κz σ s(z)z V (z) v = κz + σ s(z)z V (z) (7-a) (7-b) (7) V (z) κ s(z) σ {z : V (z) =} z = z = V (z()) = σ>κ n =3 V (z) =z + z s(z) =z 3 z z (73-a) (73-b) σ >κ κ, σ v = κz σ s(z)z V (z) v = κz + σ s(z)z V (z) (74-a) (74-b) s(z),v(z) (74) V (z) κ s(z) σ {z; V (z) =} z = z = V (z()) = σ>κ (Quasi-continuous) 44 Astolfi Astolfi [][] z = v = kz z 3 v = F z + F 3 z (75-a) (75-b) k > z F,F 3 ( ) F F 3 (76) k k z,z 3 z = open-loop σ-process χ = [ χ,χ,,χ n ], χ = z χ = χ 3 = z 3 z χ n = z n z n χ χ = v χ = v χ χ 3 χ 3 = v χ χ n = χ n (n )χ n v χ (77) (78) k> v = kχ χ = Aχ + bv (79) k A = k k, b = k k 3 (79) v = FZ (8) 9

10 Almost eponential stabilit, z () = σ-process v = kχ = kz Z v = const z () = open-loop χ = z z 3 (8) z α R n i =,,n j =,,n χ i = z i+ i +)! zi z χ(z) := [ χ χ n ] T A i, (z ) = A i,j (z ) = A(z ):= i+ (i +)! k zi+ k= { (j )! (i +)! j! A, A,n A n, A n,n (i + j)! } z i+j α(z) =λa(z ) χ(z) (86) v = kχ v = F [ ] χ χ (8-a) (8-b) α z R n [] (85) α(z(t)) = α α = α(z()) [] Astolfi z = v = λz v = λz + αz (83-a) (83-b) λ α α = λ ( ) z3 z (84) z z z() α z() z 3 α α α v Astolfi λ> v = λz v = λz + αẑ ẑ = [ z z ] T zn (85-a) (85-b) z () = n =3q, A α q = z 3 z z A = 4 z α = λa q = λ ( ) z3 z z z v = λz v = λz + αz A(z ) α = α = A(z ) (85) λ z det A(z ) α = α(z(t)) = α(z()) (86) A(z ) α 45

11 45 z ,, [4] z (t) [7] 44 ( z =) v = kd(t) F (z d(t)) v = F (t)z + F 3 (t) z 3 d(t) (87-a) (87-b) F >k> d(t) z d(t) =ι(z())e kt (88) z = z () ι(z()) ι( ) z d z z = k d F z d v z d(t) F (t),f 3 (t) Astolfi z d(t) v = kd(t) F (z d(t)) v = F (t)z + F 3 (t) z 3 d(t) + z n + F n (t) d(t) n (89-a) (89-b) F >k> d(t) z d(t) =h(z())e kt (9) F (t) =[F,,F n ](t) Astolfi F α >, z(t) k(z())e αt Astolfi z d(t) F (t) z() d() z () F >k 5 MATX[8] Runge-Kutta [msec] Fig Fig 3 5% chained form chained form chained form Fig 4,, θ Sordalen Samson chained form chained form Sordalen, Samson Astolfi, Khennouf and Canudas de Wit Pomet θ (38-a) R r R r (38-a)

12 ( ) ( ) ( ) ( ) d z3 z + = + µ dz z + (9) (39)(4) µ = k z k 3 z 3 + K z dz (z )(9) µ = k z k 3 z 3 + K z dz (z ) (93) 6 chained form Brocket [3][7][9] [][] Fig -i, Fig 3-i, Fig 4-i Fig 3-a Fig 3- i (38-a) [9] α Table : Sordalen K Samson Pomet (Lapunov) Khennouf Astolfi K {z =}

13 Figure -a: Figure 3-a: Figure -b: Sordalen Figure 3-b: Sordalen Figure -c: Samson Figure 3-c: Samson Figure -d: Pomet Figure 3-d: Pomet Figure -e: Khennouf and Canudas de Wit Figure 3-e: Khennouf and Canudas de Wit Figure -f: Astolfi Figure 3-f: Astolfi Figure -g: Figure 3-g: Figure -h: Figure 3-h: Figure -i: Figure 3-i: Figure : () Figure 3: 3

14 References [] A Astolfi Eponential stabilization of a car-like vehicle In International Conference on IEEE Robotics and Automation, pages , 995 Figure 4-a: Figure 4-b: Sordalen ~ 4 3 Figure 4-c: Samson Figure 4-d: Pomet Figure 4-e: Khennouf and Canudas de Wit Figure 4-f: Astolfi Figure 4-g: diverge Figure 4-h: 3 Figure 4-i: Figure 4: [] A Astolfi Eponential stabilization of nonholonomic sstems via discontinuous control In ProcofNOL- COS 95, pages , 995 [3] RW Brockett Asmptotic stabilit and feedback stabilization In Differential Geometric Control Theor, volume 7, pages 8 9 Springer Verlag, 983 [4] J Imura, K Kobaashi, and T Yoshikawa Eponential stabilization problem of nonholonomic chained sstem with specified transient response In Proc of the 35th CDC, pages , 996 [5] H Khennouf and C Canudas de Wit On the construction of stabilizing discontinuous controllers for nonholonomic sstems In Proc of NOLCOS 95, pages , 995 [6] H Khennouf and C Canudas de Wit Quasicontinuous eponential stabilizers for nonholonomic sstems In IFAC 3th World Congress, pages b 7 4, San Francisco,USA, 996 International Federation of Automatic Control [7] H Kiota and M Sampei A control of a class of nonholonomic sstems with drift using time-state control form In Proc of the th SICE smp on Dnamical Sstem Theor, pages 9 3, 997 [8] RM Murra and SS Sastr Nonholonomic motion planning: Steering using sinusoids IEEE Trans on Automatic Control, 38(5):7 76, 993 [9] T Nakagawa, H Kiota, M Sampei, and M Koga An adaptive control of a nonholonomic space robot In Proc of the 36th IEEE Conference on Decision and Control, pages , 997 [] J-B Pomet Eplicit design of time-varing stabilizing control laws for a class of controllable sstems without drift Sstems & Control Letters, 8:47 58, 99 [] M Sampei A control strateg for a class fo nonholonomic sstems time-state control form and its application In Procof33rdCDC, pages, 994 [] M Sampei, H Kiota, M Koga, and M Suzuki Necessar and sufficient conditions for transformation of nonholonomic sstem into time-state control form In 996 IEEE Conference on Decision and Control, pages , 996 [3] M Sampei, H Kiota, S Mizuno, and M Koga A control of a class of nonholonomic sstems subject to velocit constraints using acceleration inputs In AACC Proc of 997 American Control Conference, pages 3 3, 997 4

15 [4] C Samson Control of chained sstems application to path following and time-varing point-stabilization of mobile robots IEEE Trans on Automatic Control, 4():64 77, 995 [5] OJ Sordalen and O Egeland Eponential stabilization of nonholonomic chained sstems IEEE Trans on Automatic Control, 4():35 49, 995 [6] P Tsiotras, M Corless, and JM Longuski Invariant manifold techniques for attitude control of smmetric spacecraft In Proc of 3nd CDC, 993 [7],, and In, pages 85 86, 995 [8] ( ) mat, 4(6):8 83, 996 [9], 36(6):44 4, 997 [],,, and, 4(8):37 4, 996 [],,, and, 6():8 3, 998 [] and Chained form, 3(8):3 3, 996 5

meiji_resume_1.PDF

meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E