Collision Avoidance Control for a Two-wheeled Vehicle under Stochastic Vibration using an Almost Sure Control Barrier Function

Collision Av oidance Con trol for a Tw o-wheeled V ehicle under Sto c hastic Vibration using an Almost Sure Con trol Barrier F unction T aic hi Arim ura ∗ , Yˆ uki Nishim ura † , T aichi Ik ezaki † and Daisuk e T abuchi ∗ Marc h 30, 2026 ‡ Abstract In recen t years, man y control problems of autonomous mobile robots ha ve been dev elop ed. In particular, the rob ots are required to b e safe; that is, they need to b e controlled to a void colliding with p eople or ob jects while trav eling. In addition, since safet y should b e ensured ev en under irregular disturbances, the control for safety is required to b e effective for sto c hastic systems. In this study , we design an almost sure safety-critical con trol la w, whic h ensures safety with probabilit y one, for a tw o-wheeled v ehicle based on the stochastic con trol barrier function approac h. In the pro cedure, we also consider a system mo del using the relative distance measured b y a 2D LiDAR. The v alidity of the proposed control scheme is confirmed b y exp erimen ts of a collision a voidance problem for a tw o- wheeled vehicle under vibration. 1 In tro duction The dev elopment of autonomous mobile rob ots, such as cleaning rob ots and automatic guided vehicles, is promoted to improv e work efficiency and solve lab or shortages [1, 2]. Because autonomous mobile rob ots are often used in residen tial spaces and factories with surrounding obstacles, they are required to b e safe enough to a void collisions with people or ob jects while tra veling. The collision a voidance problems are solved via v arious methods: the artifi- cial p oten tial field metho d by Rimon and Koditschek [3], the dynamic windo w approac h by F o x, Burgard and Thrun[4] and Hahn[5], the nearness diagram metho d by Minguez and Montano [6], a na vigation metho d considering moving obstacles by Tsub ouchi et al. [7], a collision a voidance metho d based on fuzzy ∗ Kagoshima Universit y † Ok a yama Univ ersity ‡ This work has b een submitted to the SICE Journal of Control, Measuremen t, Systems and Integration for possible publication. Copyrigh t ma y b e transferred without notice, after which this v ersion ma y no longer b e accessible. 1 inference b y Maeda and T akegaki [8], and a navigation metho d considering the future b eha vior of dynamic obstacles by Z. Zhang et al. [9]. Recen tly , a safet y-critical control approach based on control barrier functions [10] is attracting attention as a scheme for maintaining the safety of mobile rob ots b ecause of its simple and strong con trol design. Kim ura and Nishimoto [11] prop ose a design metho d for collision av oidance con trol of a tw o-wheeled v ehicle using measured data of the distance b etw een the obstacle and the vehicle via 2D LiDAR based on [12, 13]. A t the same time, the actual tra veling en vironment of mobile rob ots is often influenced b y irregular disturbances. Therefore, designing a con trol law that ac hieves the safet y-critical con trol ob jective even under the influence of irregular disturbances is required. More clearly , the safety is preferable to hold with 100%; that is, almost sur ely , even when white noise vibrates the robots. Nishimura and Hoshino [14] propose a design pro cedure for an almost sur e safety-critic al c ontr ol law for nonlinear sto chastic systems including Gaussian white noise. Therefore, the combination of the metho ds in [11] and [14] is effectiv e for controlling a t wo-wheeled v ehicle in the real-world en vironment. In this pap er, we apply the almost sure safet y-critical control scheme by Nishim ura and Hoshino [14] to the collision av oidance system of a tw o-wheeled v ehicle b y Kim ura and Nishimoto [11], and then deriv e a con trol strategy that ac hieves collision a voidance with 100% against the existence of irregular distur- bances. The organization of this pap er is as follows. In Section 2, we briefly sum- marize the previous works used in this study . In Section 3, w e present the sto c hastic system of the tw o-wheeled vehicle considered as the con trol target. In Section 4, we describe the almost sure safety con trol law for a v oiding colli- sions b et ween the v ehicle and obstacles. In Section 5, w e confirm and discuss the v alidity of the prop osed collision av oidance controller via simulations and exp erimen ts. Finally , in Section 6, we conclude this pap er. Notation. W e in tro duce the notation used throughout this pap er. R n denotes the n -dimensional Euclidean space and esp ecially R := R 1 . R ≥ 0 also denotes the set of nonnegative real num b ers. The Lie deriv ative of a smo oth mapping W : R n → R along a mapping F = ( F 1 , . . . , F q ) : R n → R n × q is defined as L F W ( x ) =  ∂ W ∂ x F 1 ( x ) , . . . , ∂ W ∂ x F q ( x )  . (1) The b oundary of a set χ is denoted by ∂ χ . The mapping w : [0 , ∞ ) → R denotes a one-dimensional standard Wiener pro cess. The differential form of the Itˆ o in tegral of a mapping σ : R n → R along w is denoted b y σ ( x ) dw . The trace of a square matrix A is denoted by tr[A]. Exp erimental Envir onment. W e use a Lightro ver made b y Vstone Co., Ltd., sho wn in Fig. 1 as a t wo-wheeled v ehicle, a YDLiDAR X2 made b y Shenzhen EAI T ec hnology Co., Ltd, as a 2D LiDAR, and a Balancew av e Rose F A V4318P made by ALINCO Inc., shown in Fig. 2 as a noise source, which is capable 2 Figure 1: Lightro ver. Figure 2: Balancew av e rose F A V4318P . of applying mixed vibration combining three-dimensional vibration and micro vibration. 2 Preliminary: Safet y-critical Con trol 2.1 Safet y-critical Con trol In this subsection, we briefly summarize a safety-critical con trol law proposed in [13]. Consider an input-affine nonlinear con trol system ˙ x ( t ) = f ( x ( t )) + g ( x ( t ))( u o ( t ) + u ( t )) , (2) where x : [0 , ∞ ) → R n is a state, u o : [0 , ∞ ) → R m is a preinput assumed to be contin uous, u : [0 , ∞ ) → R m is a comp ensator for safet y-critical con trol, f : R n → R n and g : R n → R n × m are b oth assumed to b e lo cally Lipschitz con tinuous, and an initial state is given as x 0 = x (0) ∈ R n . Consider an op en set χ ⊂ R n and a contin uously differentiable function B : χ → R . If the following assumptions: (A1) B ( x ) ≥ 0 for all x ∈ χ , (A2) for any L ≥ 0, { x ∈ χ | B ( x ) ≤ L } is compact, (A3d) for any contin uous mapping u o : R → R m , there exist nonnegative constan ts C , K ≥ 0 suc h that inf u ∈ R m ˙ B ( x, u o , u ) < K B ( x ) + C, (3) are all satisfied, then χ and B ( x ) are said to be a safe set and a c ontr ol b arrier function (CBF) , resp ectiv ely . Under the assumptions (A1), (A2) and (A3d), designing u = ϕ D ( t, x ) with ϕ D ( t, x ) = ( − I d ( x,u o ) − J d ( x ) || L g B ( x ) || 2 ( L g B ( x )) T , I d ( x, u o ) > J d ( x ) , 0 , otherwise , (4) 3 where I d ( x, u o ) = L f B ( x ) + L g B ( x ) u o , (5) J d ( x ) = K B ( x ) + C , (6) the system (2) is safe in χ ; that is, the tra jectory x k eeps sta ying in χ for any initial v alue x 0 ∈ χ . 2.2 Almost Sure Safet y-critical Con trol In this subsection, we consider an almost sure safety-critical con trol law that k eeps the state in the safe set with probabilit y one, prop osed in [14]. Assuming that the system (2) is influenced by Gaussian white noise, w e obtain a sto c hastic system dx ( t ) = { f ( x ( t )) + g ( x ( t ))( u o ( t ) + u ( t )) } dt + σ ( x ( t )) dw ( t ) , (7) where f , g , u o , u, x and x 0 are the same as those in (2), and σ : R n → R n × d is assumed to b e locally Lipschitz con tinuous. Let L I σ ( y ( x )) := 1 2 tr " σ ( x ) σ ( x ) T " ∂ ∂ x  ∂ y ∂ x  T # ( x ) # , (8) L f ,g,σ ( u, u o , y ( x )) := ( L f y )( x ) + ( L g y )( x )( u + u o ) + L I σ ( y ( x )) , (9) where B : χ → R is twice con tinuously differentiable, and an op en set χ ⊂ R n . If (A1), (A2) and (A3s) for an y con tinuous mapping u o : R → R m , there exist nonnegative constan ts γ ≥ 0 such that inf u ∈ R m L f ,g,σ ( u, u o , B ( x )) ≤ γ B ( x ) , (10) are all satisfied, then χ and B ( x ) are said to b e a safe set and a almost sur e r e cipr o c al c ontr ol b arrier function (AS-RCBF) , respectively . Under the assumptions (A1), (A2) and (A3s), design ϕ N ( t, x ) =  ψ ( t, x ) , I > J ∩ L g B  = 0 , 0 , I ≤ J ∪ L g B = 0 , (11) where I ( u o , B ( x )) := L f ,g,σ (0 , u o , B ( x )) , (12) J ( B ( x )) := γ B ( x ) , (13) ψ ( t, x ) := − I ( u o , B ( x )) − J ( B ( x )) L g B ( x ) L g B ( x ) T L g B ( x ) T . (14) 4 If L f ( B ( x )) + L I σ ( B ( x )) > γ B ( x ) (15) holds on L g h = 0, then the compensator u = ϕ N ( t, x ) is contin uous and makes the system (7) safe in χ with probability one; that is, the tra jectory x keeps sta ying in χ for any initial v alue x 0 ∈ χ with 100% probabilit y [14]. In addition, if σ = 0 for all x , ϕ N = ϕ D with K = γ and C = 0. Remark 1 In this p ap er, the pr einput u o is assume d to b e time varying; that is, u o ( t ) , while it is assume d to b e time invariant; that is, u o ( x ( t )) in [14]. In fact, the r esults fr om [14] ar e dir e ctly applie d to u o ( t ) b e c ause the r esults ar e b ase d on the forwar d invarianc e in pr ob ability (FIiP), a sufficient c ondition for which is given in [15]. This r emark is also state d in [16], and the same r esult on deterministic systems is describ e d in [13] as shown in Subse ction 2.1. 3 T arget System 3.1 Dynamics for a V ehicle with 2D LiDAR In this subsection, we consider the dynamics of the t wo-wheeled v ehicle, sho wn in Fig. 3, based on [11]; the difference b et ween our vehicle and the v ehicle in [11] will b e stated in Remark 2 at the end of this section. W e define the translational velocity v o + v and the rotational v elo cit y w o + w as inputs, resp ectiv ely , where v o and w o are preinputs (previously giv en inputs), and v and w are comp ensators for safety-critical control. Adding the inputs for the vehicle, w e obtain ˙ p 1 = ( v o + v ) cos p 3 , (16) ˙ p 2 = ( v o + v ) sin p 3 , (17) ˙ p 3 = ( w o + w ) , (18) where p 1 and p 2 describ es the center p osition B of the axle, and p 3 as the angular difference from the p 1 axis. W e obtain the distance from the v ehicle to an obstacle using the 2D LiDAR moun ted on the vehicle, where the forw ard direction is defined as 0 rad, the measuremen t range is 2 π rad, and the n umber of measuremen t points is denoted b y N . F or i ∈ { 1 , . . . , N } , the distance and the angle of the i -th p oin t are denoted b y x 1 i ∈ R ≥ 0 and x 2 i ∈ [ − π , π ), resp ectiv ely . Set the time sequence for measuring as t 0 , t 1 , . . . , t k with t 0 = 0, the time in terv als as [ t k , t k +1 ) for any k = 1 , 2 , . . . , and assume that t k +1 − t k is a constant for an y k = 1 , 2 , . . . . The absolute co ordinates of the i -th p oin t observed at time t = t k with k ∈ { 1 , 2 , . . . } are represented b y ¯ x i,k =  ¯ x 1 i,k ¯ x 2 i,k  =  x 1 i cos( x 2 i + p 3 ) + p 1 x 1 i sin( x 2 i + p 3 ) + p 2  . (19) 5 Figure 3: A system mo del of a tw o- wheeled vehicle. Figure 4: Relationships among con- stan ts and v ariables. Assuming that the obstacle is time-in v ariant and the vehicle acts sufficien tly slo wly , ¯ x i,k is constan t. Hence, differen tiating (19), using ˙ ¯ x 1 ,k = ˙ ¯ x 2 ,k = 0 and defining x i = [ x 1 i , x 2 i ] T , we obtain ˙ x i = g i ( x )  v o + v w o + w  , g i ( x ) =  − cos x 2 i 0 sin x 2 i x 1 i − 1  , (20) for i = 1 , . . . , N in t ∈ [ t k , t k +1 ). Since w e will consider t ∈ [ t k , t k +1 ) hereafter, w e omit the subscript k for simplicity notation, as in [11]. Setting x =  x T 1 , . . . , x T i , . . . , x T N  T , g ( x ) =  g T 1 ( x 1 ) . . . g T i ( x i ) . . . g T N ( x N )  T , (21) w e obtain the follo wing system mo del: ˙ x = g ( x )( u o + u ) , (22) where x ∈ R n with n = 2 N . 3.2 Distance from the Sensor to the Axle In this subsection, w e derive the distance and the angle from the cen ter of the driving wheels based on the measurement results b y the sensor. In our vehicle sho wn in Fig. 4, the center of the sensor S is not equiv alent to the cen ter of the driving wheels B b ecause of the sp ecifications. This implies that we ha ve to clarify the relationship b et ween S and B . Let d b e the distance b et ween S and B . F or i ∈ { 1 , 2 , . . . , N } , x S 1 i and x S 2 i are the distance and the angle b etw een the measured p oin t x i and the 6 sensor, resp ectively . Then, considering S as the origin, x i is expressed as ( x S 1 i cos x S 2 i , x S 1 i sin x S 2 i ). Therefore, x 1 i and x 2 i are given b y x 1 i = p ( x S 1 i cos x S 2 i − d ) 2 + ( x S 1 i sin x S 2 i ) 2 , (23) x 2 i = arccos x S 1 i cos x S 2 i − d x 1 i , (24) resp ectiv ely . 3.3 Allo w able Distance from the Axle to an Obstacle In this subsection, we consider the allow able distance to an obstacle. W e set our control ob jective to keep the distance betw een O and an ob ject within the desired v alue α > 0. Therefore, we consider the threshold α ci of the distance b et w een B and an obstacle. Denoting e as the distance b et ween O and B , α ci = p ( α cos θ − e ) 2 + ( α sin θ ) 2 , (25) tan x 2 i = α sin θ α cos θ − e , (26) α sin θ = α ci sin x 2 i . (27) are obtained. Thus, assuming α > e > 0, α ci is represen ted as the function of x 2 i : α ci = − e cos x 2 i + q α 2 − e 2 sin 2 x 2 i . (28) 3.4 Sto c hastic System Mo del In this subsection, we consider the situation where noise is added to our system mo del. Letting σ i = [ c 1 , c 2 ] T and assuming that a Gaussian white noise is added in to the system (20), w e obtain a sto c hastic system dx i = g i ( x i )( u o + u ) dt + σ i dw (29) for each i ∈ { 1 , 2 , . . . , N } . Then, defining σ =  σ T 1 , . . . , σ T i , . . . , σ T N  T , (30) w e obtain our target system dx = g ( x )( u o + u ) dt + σ dw , (31) whic h is a stochastic v ersion of (22). Remark 2 Our vehicle mo del in (31) has differ enc es fr om the vehicle in [11]. First, in our vehicle, the c enter B of the driving whe els is not e quivalent to the sensor p osition S . Se c ond, our vehicle is cir cular. Thir d, we c onsider the angle x 2 i as a state variable of the system mo del to ensur e c onsistency with the the ory. A nd final ly, we assume the system is vibr ate d by sto chastic noise. 7 4 Almost Sure Safet y-critical Con trol La w In this section, we prop ose an AS-R CBF for the system (31) and design an almost sure safet y control law to av oid collisions b etw een the vehicle and ob- stacles based on the design pro cedure of a CBF in [11] and the almost sure safet y-critical control theory in Section 2.2. F or the i -th p oint x i , we define χ i = { x i | x 1 i > α ci } , (32) B i ( x i ) = 1 x 1 i − α ci . (33) Because our goal is to keep x 1 i > α ci for all i , we define χ = N Y i =1 χ i , (34) B ( x ) = N X i =1 B i ( x i ) , (35) as a safe set and an AS-RCBF for (31), resp ectiv ely . Then, ψ ( x ) in (14), which is the part of the compensator ϕ N , results in ψ ( t, x ) = −  v o w o  + γ B ( x ) − L I σ ( B ( x )) || L g B ( x ) || 2 ( L g B ( x )) T , (36) where L g B ( x ) = N X i =1 L g i B i = N X i =1 1 ( x 1 i − α ci ) 2  cos x 2 i + sin x 2 i x 1 i α ′ ci − α ′ ci  T , (37) L I σ ( x ) = 1 2 N X i =1 σ T i ∂ ∂ x i  ∂ B i ∂ x i  T σ i = N X i =1 1 ( x 1 i − α ci ) 3 σ T i  1 − α ′ ci − α ′ ci β i  σ i , (38) α ′ ci := ∂ α ci ∂ x 2 i = e sin x 2 i − e 2 sin ( x 2 i ) cos ( x 2 i ) p α 2 − e 2 sin 2 x 2 , (39) β i := α ′ ci  α ′ ci + 1 2 ( x 1 i − α ci ) ∂ α ′ ci ∂ x 2 i  . (40) Remark 3 While B ( x ) in (35) do es not satisfy (A2), the c ondition is mer ely a the or etic al r e quir ement; that is, if (A2) is satisfie d, the existenc e of a glob al solution in time is ensur e d. We c an design a the or etic al ly ac cur ate AS-R CBF as ¯ B ( x ) = B ( x ) + εP ( x ) , (41) wher e P : χ → [0 , ∞ ) is a design function such that it is c omp act in { x ∈ χ | P ( x ) ≤ L } for any L > 0 , and ε > 0 is a design p ar ameter. Be c ause we c an cho ose ε to b e arbitr arily smal l, we ignor e the term εP ( x ) when de aling with c ontr ol issues in physic al e quipment. 8 Figure 5: Environmen t of Exps. 1d and 1n. Figure 6: Environmen t of Exps. 2d and 2n. 5 Exp erimen ts 5.1 P arameter Settings W e describ e the parameter v alues used in the exp erimen ts. In the exp erimen tal environmen t, the num b er of the measurement points is N = 279, the distance b et ween the center of the sensor S and the center of the driving wheels B is d = 0 . 07 m, and the distance b et ween the cen ter of the v ehicle O and B is e = 0 . 025 m. W e design parameters as γ = 0 . 5 and α = 0 . 3, and determine the diffusion co efficien ts as c 1 = 0 . 035 and c 2 = 0 using the estimation procedure shown in App endix A. W e also consider the deterministic controller u = ϕ D for compar- ison, which is describ ed in (4) with K = γ = 0 . 5 and C = 0, and derived by assuming c 1 = c 2 = 0 in (36). Then, w e set preinputs to v o = 0 . 2 and w o = 0 . 2, respectively , and the the initial state x 1 i ∗ (0) = 0 . 4 and x 2 i ∗ (0) = π / 4, where i ∗ = i ∗ ( t ) is the subscript suc h that satisfies x 1 i ∗ − α ci ∗ = min i ∈{ 1 , 2 ,...,N } ( x 1 i − α ci ). 5.2 Sim ulation and Exp erimental Results First, we set the v ehicle on the floor as sho wn in Fig. 5 and perform numerical sim ulations and conduct exp erimen ts with the following conditions, respectively: (Exp. 1d) u = ϕ D ; (Exp. 1n) u = ϕ N with c 1 = 0 . 035. Then, we set the vehicle on the vibration platform as shown in Fig. 6. and p erform n umerical sim ulations and conduct exp eriments with the following con- ditions, resp ectiv ely: (Exp. 2d) u = ϕ D ; (Exp. 2n) u = ϕ N with c 1 = 0 . 035. 9 Figure 7: T ra jectories of the cen ter O of the vehicle in Exps. 1d ( ϕ D ) and 1n ( ϕ N ). Figure 8: T ra jectories of the center O of the v ehicle in Exps. 1d ( ϕ D ) and 1n ( ϕ N ). The tra jectories of Exps. 1d and 1n are sho wn in Fig. 7, and Exps. 2d and 2n, Fig. 8, respectively . The time responses of the states, the inputs, and the AS-RCBF B ( x ) of Exps. 1d, 1n, 2d, and 2n are sho wn in Figs. 9–18 and Figs. 19–28, respectively; the green and y ellow lines sho w the time responses and the boundary α ci ∗ of the n umerical sim ulations, resp ectiv ely . The blue and red lines also sho w the time responses and α ci ∗ from ten trials of the exp erimen ts, resp ectiv ely . 5.3 Discussion In this subsection, we confirm the v alidity of the prop osed controller via discus- sions on the results of the simulation and experimental results. First, we compare the collision-av oiding results without adding noise. In Fig. 9 (Exp. 1d), the deterministic safety-critical control u = ϕ D is effectiv e for a voiding collision to the threshold α ci ∗ in sim ulation results, while x 1 i ∗ mo ved sligh tly past α ci ∗ to ward the wall. In con trast, in Fig. 14 (Exp. 1n), the almost sure safety-critical con trol u = ϕ N successfully av oided the collision. The dif- ference is also shown in the tra jectories sho wn in Fig. 7. W e consider that the cause of the failure in Exp. 1d is the existence of mo deling and measuremen t errors, as well as v ariations in sample timing during actual measurements, and the cause of the success in Exp. 1n is that the safet y comp ensation for noise is also effective for atten uating the errors and the v ariations. Second, w e compare the collision-av oiding results with adding noise. Com- paring the tra jectories in Fig. 8, or, time resp onses of x i ∗ in Figs. 19 (Exp. 2d) and 24 (Exp. 2n), the almost sure safet y-critical control u = ϕ N is effective for collision av oidance, while the deterministic safety-critical con trol u = ϕ D fails the collision av oidance. Third, w e compare the time resp onses of AS-R CBF B ( x ). In Fig. 13 (Exp. 1d) and Fig. 23 (Exp. 1n), the v alue of B ( x ) rapidly b ecomes a large v alue as B − 1 i ∗ = x 1 i ∗ − α ci ∗ , and thereafter it b ecomes negativ e. While B ( x ) diverges as B i ∗ reac hes zero theoretically , it is considered that the v alue remained finite due to a measurement error in actual measurement, and then, B i ∗ b ecomes nega- tiv e in the subsequent time step. In contrast, in Fig. 18 (Exp. 1n) and Fig.28 10 (Exp. 2n), B ( x ) k eeps p ositiv e finite v alue; it implies that the safet y is ensured with 100%. Finally , we consider rapid c hanges in input v alues. As in Figs. 11, 12, 16, 17, 21, 22, 26 and 27, the mov ements of v o + v and w o + w exhibit vibrations not caused b y noise. The phenomena occur when the angle x 2 i ∗ ≈ − π / 2 as sho wn in Figs. 10, 15, 20 and 15; that is, the v ehicle is facing parallel to the w all. Viewing (37), the signs of the elements of L g B b oth changes around x 2 i ∗ = − π / 2; this implies that the signs of v o + v and w o + w change frequen tly when x 2 i ∗ ≈ − π / 2. Therefore, we estimate that the phenomena are due to the form of the prop osed con trol law. T o attenuate the mov ements, an appropriate design strategy for an AS-R CBF will b e required. Otherwise, using a LiD AR with an extremely high n umber of measurement p oints, eac h elemen t of L g i B i with a positive or negativ e sign is canceled out in the calculation of L g B . The atten uation of the phenomena will b e an important issue of future works. 6 Conclusion In this pap er, we designed an almost sure safet y-critical control la w based on an almost sure recipro cal control barrier function for sto c hastic systems prop osed b y Nishimura and Hoshino [14], using the state equation of the relative distance based on Kimura and Nishimoto [11]. W e confirmed the v alidity of the designed con trol law that ac hieves collision av oidance via numerical sim ulations and we exp erimen ts. As a result, when the con trol law was designed without assuming noise, the vehicle crossed the b oundary and safety was not main tained. In con trast, by incorp orating noise into the con trol design, we ac hieved collision a voidance of the vehicle and main tained its safet y . These results demonstrated that the almost sure safet y-critical con trol la w is effective even under sto chastic vibration. Ac kno wledgemen t(s) W e would lik e to express our gratitude to Mr. T akumi Y amaok a at Kagoshima Univ ersity and Mr. Kazuma Miy ay oshi at Ok ay ama Universit y for their coop- eration in conducting the exp erimen ts for this pap er. References [1] E. T akeuc hi. Developmen t of mobile rob ots using middleware for rob ots, Journal of the So ciety of Instrument and Contr ol Engine ers V ol. 57, No. 10, pp. 741–744, 2018. [2] K. Maek aw a. Autonomous mobile rob ot for factory automation and logistics automation, Systems, Contr ol and Information V ol. 64, No. 5, pp. 177–181, 2020. 11 [3] E. Rimon, D.E. Ko ditsc hek. 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Almost sure front collision prev ention control for an electric wheelchair via sto chastic safety- critical control theory , T r ansactions of the Institute of Systems, Contr ol and Information Engine ers , accepted in F ebruary 2026. A Estimation of Diffusion Co efficien ts In this section, we estimate the v alues of the diffusion coefficients. F or simplicit y , w e assume c 2 = 0 and estimate c 1 from preliminary vibration exp erimen ts on a vibration platform. W e set x 2 i = 0 and v o + v = 0; that is, dx 1 i = c 1 dw , (42) and then, by applying the Euler-Maruyama scheme, we obtain the following discrete-time system x 1 i ( t j +1 ) = x 1 i ( t j ) + v ( t j )∆ t + c 1 ∆ w j , (43) where j ∈ { 0 , 1 , 2 , . . . } , ∆ t := t j +1 − t j is constant for an y j , and c 1 ∆ w j = x 1 i ( t j +1 ) − x 1 i ( t j ) − v ( t j )∆ t. (44) Th us, the v ariance of c 1 ∆ w j is calculated as V ar( c 1 ∆ w j ) = V ar ( x 1 i ( t j +1 ) − x 1 i ( t j ) − v ( t j )∆ t ) (45) = c 2 1 ∆ t, (46) where V ar( A ) is the v ariance of A . Therefore, the noise coefficient c 1 is deriv ed as c 1 = r V ar ( x 1 i ( t j +1 ) − x 1 i ( t j ) − v ( t j )∆ t ) ∆ t . (47) Then, w e identify the diffusion co efficien t c 1 b y conducting preliminary ex- p erimen ts. W e place the v ehicle on a vibration platform, actually vibrate it, and measure the noise ten times. The experiments result in V ar( c 1 ∆ w j ) = 0 . 00012 and ∆ t = 0 . 1, and then, the v alue of c 1 is iden tified as c 1 = r 0 . 00012 0 . 1 ≈ 0 . 035 . (48) 13 Figure 9: Time resp onses of x 1 i ∗ and α ci ∗ in Exp. 1d. Figure 10: Time resp onses of x 2 i ∗ in Exp. 1d. Figure 11: Time responses of v o + v in Exp. 1d. Figure 12: Time resp onses of w o + w in Exp. 1d. Figure 13: Time resp onses of B ( x ) in Exp. 1d. Figure 14: Time responses of x 1 i ∗ and α ci ∗ in Exp. 1n. Figure 15: Time resp onses of x 2 i ∗ in Exp. 1n. Figure 16: Time responses of v o + v in Exp. 1n. Figure 17: Time resp onses of w o + w in Exp. 1n. Figure 18: Time resp onses of B ( x ) in Exp. 1n. 14 Figure 19: Time responses of x 1 i ∗ and α ci ∗ in Exp. 2d. Figure 20: Time resp onses of x 2 i ∗ in Exp. 2d. Figure 21: Time responses of v o + v in Exp. 2d. Figure 22: Time resp onses of w o + w in Exp. 2d. Figure 23: Time resp onses of B ( x ) in Exp. 2d. Figure 24: Time responses of x 1 i ∗ and α ci ∗ in Exp. 2n. Figure 25: Time resp onses of x 2 i ∗ in Exp. 2n. Figure 26: Time responses of v o + v in Exp. 2n. Figure 27: Time resp onses of w o + w in Exp. 2n. Figure 28: Time resp onses of B ( x ) in Exp. 2n. 15